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! # !! Formula !! Title !! Evaluation data | ! # !! Formula !! Title !! Evaluation data | ||
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| 1 | | 1 | ||
| <syntaxhighlight lang="tex" inline >\begin{align}J_{-(m+\frac{1}{2})}(x) &= (-1)^{m+1} Y_{m+\frac{1}{2}}(x), \\Y_{-(m+\frac{1}{2})}(x) &= (-1)^m J_{m+\frac{1}{2}}(x).\end{align}</syntaxhighlight> | |||
| [[Bessel function#math.51.18| Bessel function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 2 | |- | ||
| 2 | |||
| <syntaxhighlight lang="tex" inline >E(e) \,=\, \int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta</syntaxhighlight> | |||
| [[Ellipse#math.52.404| Ellipse]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 3 | |- | ||
| 3 | |||
| <syntaxhighlight lang="tex" inline >F(x;k) = u</syntaxhighlight> | |||
| [[Elliptic integral#math.53.6| Elliptic integral]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 4 | |- | ||
| 4 | |||
| <syntaxhighlight lang="tex" inline >\frac{1}{\Gamma(z)} = \frac{i}{2\pi}\int_C (-t)^{-z}e^{-t}\,dt</syntaxhighlight> | |||
| [[Gamma function#math.54.195| Gamma function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 5 | |- | ||
| 5 | |||
| <syntaxhighlight lang="tex" inline >2^{4} = 2 \times2 \times 2 \times 2 = 16</syntaxhighlight> | |||
| [[Logarithm#| Logarithm]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 6 | |- | ||
| 6 | |||
| <syntaxhighlight lang="tex" inline >\psi(x) := \sum_{n=1}^\infty e^{-n^2 \pi x}</syntaxhighlight> | |||
| [[Riemann zeta function#math.56.40| Riemann zeta function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 6, | "id": 6, | ||
"pid": 56, | "pid": 56, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 7 | |- | ||
| 7 | |||
| <syntaxhighlight lang="tex" inline >\operatorname{li}(x) = \lim_{\varepsilon \to 0+} \left( \int_0^{1-\varepsilon} \frac{dt}{\ln t} + \int_{1+\varepsilon}^x \frac{dt}{\ln t} \right)</syntaxhighlight> | |||
| [[Logarithmic integral function#math.57.2| Logarithmic integral function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 7, | "id": 7, | ||
"pid": 57, | "pid": 57, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 8 | |- | ||
| 8 | |||
| <syntaxhighlight lang="tex" inline >w_{i} = \frac{1}{p'_{n}(x_{i})}\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx</syntaxhighlight> | |||
| [[Gaussian quadrature#math.58.61| Gaussian quadrature]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 8, | "id": 8, | ||
"pid": 58, | "pid": 58, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 9 | |- | ||
| 9 | |||
| <syntaxhighlight lang="tex" inline >\begin{align}x & =ue^u, \\[5pt]\frac{dx}{du} & =(u+1)e^u.\end{align}</syntaxhighlight> | |||
| [[Lambert W function#math.59.52| Lambert W function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 9, | "id": 9, | ||
"pid": 59, | "pid": 59, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 10 | |- | ||
| 10 | |||
| <syntaxhighlight lang="tex" inline >\frac{1}{\left| \mathbf{x}-\mathbf{x}' \right|} = \frac{1}{\sqrt{r^2+{r'}^2-2r{r'}\cos\gamma}} = \sum_{\ell=0}^\infty \frac{{r'}^\ell}{r^{\ell+1}} P_\ell(\cos \gamma)</syntaxhighlight> | |||
| [[Legendre polynomials#math.60.57| Legendre polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 10, | "id": 10, | ||
"pid": 60, | "pid": 60, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 11 | |- | ||
| 11 | |||
| <syntaxhighlight lang="tex" inline >\operatorname{erf}^{(k)}(z) = \frac{2 (-1)^{k-1}}{\sqrt{\pi}} \mathit{H}_{k-1}(z) e^{-z^2} = \frac{2}{\sqrt{\pi}} \frac{d^{k-1}}{dz^{k-1}} \left(e^{-z^2}\right),\qquad k=1, 2, \dots</syntaxhighlight> | |||
| [[Error function#math.61.27| Error function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 11, | "id": 11, | ||
"pid": 61, | "pid": 61, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 12 | |- | ||
| 12 | |||
| <syntaxhighlight lang="tex" inline >x_k = \cos\left(\frac{\pi(k+1/2)}{n}\right),\quad k=0,\ldots,n-1</syntaxhighlight> | |||
| [[Chebyshev polynomials#math.62.44| Chebyshev polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 12, | "id": 12, | ||
"pid": 62, | "pid": 62, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 13 | |- | ||
| 13 | |||
| <syntaxhighlight lang="tex" inline >E(x, y; u) := \sum_{n=0}^\infty u^n \, \psi_n (x) \, \psi_n (y) = \frac{1}{\sqrt{\pi (1 - u^2)}} \, \exp\left(-\frac{1 - u}{1 + u} \, \frac{(x + y)^2}{4} - \frac{1 + u}{1 - u} \, \frac{(x - y)^2}{4}\right)</syntaxhighlight> | |||
| [[Hermite polynomials#math.63.109| Hermite polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 13, | "id": 13, | ||
"pid": 63, | "pid": 63, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 14 | |- | ||
| 14 | |||
| <syntaxhighlight lang="tex" inline >x = \pm 1</syntaxhighlight> | |||
| [[Legendre function#math.64.8| Legendre function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 14, | "id": 14, | ||
"pid": 64, | "pid": 64, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 15 | |- | ||
| 15 | |||
| <syntaxhighlight lang="tex" inline >E_n=2^nE_n(\tfrac{1}{2})</syntaxhighlight> | |||
| [[Bernoulli polynomials#math.65.27| Bernoulli polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 15, | "id": 15, | ||
"pid": 65, | "pid": 65, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 16 | |- | ||
| 16 | |||
| <syntaxhighlight lang="tex" inline >\operatorname{Si}(ix) = i\operatorname{Shi}(x)</syntaxhighlight> | |||
| [[Trigonometric integral#math.66.8| Trigonometric integral]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 16, | "id": 16, | ||
"pid": 66, | "pid": 66, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 17 | |- | ||
| 17 | |||
| <syntaxhighlight lang="tex" inline >f(z)=\frac{1}{\Beta(x,y)}</syntaxhighlight> | |||
| [[Beta function#math.67.29| Beta function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 17, | "id": 17, | ||
"pid": 67, | "pid": 67, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 18 | |- | ||
| 18 | |||
| <syntaxhighlight lang="tex" inline >\begin{align}\int x^m e^{ix^n}\,dx & =\frac{x^{m+1}}{m+1}\,_1F_1\left(\begin{array}{c} \frac{m+1}{n}\\1+\frac{m+1}{n}\end{array}\mid ix^n\right) \\[6px]& =\frac{1}{n} i^\frac{m+1}{n}\gamma\left(\frac{m+1}{n},-ix^n\right),\end{align}</syntaxhighlight> | |||
| [[Fresnel integral#math.68.51| Fresnel integral]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 18, | "id": 18, | ||
"pid": 68, | "pid": 68, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 19 | |- | ||
| 19 | |||
| <syntaxhighlight lang="tex" inline >T_n(x) = \frac{\Gamma(1/2)\sqrt{1-x^2}}{(-2)^n\,\Gamma(n+1/2)} \ \frac{d^n}{dx^n}\left([1-x^2]^{n-1/2}\right)</syntaxhighlight> | |||
| [[Classical orthogonal polynomials#math.69.117| Classical orthogonal polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 19, | "id": 19, | ||
"pid": 69, | "pid": 69, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 20 | |- | ||
| 20 | |||
| <syntaxhighlight lang="tex" inline >{}_1F_0(1;;z) = \sum_{n \geqslant 0} z^n = (1-z)^{-1}</syntaxhighlight> | |||
| [[Generalized hypergeometric function#math.70.58| Generalized hypergeometric function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 20, | "id": 20, | ||
"pid": 70, | "pid": 70, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 21 | |- | ||
| 21 | |||
| <syntaxhighlight lang="tex" inline >\chi(-1) = 1</syntaxhighlight> | |||
| [[Dirichlet L-function#math.71.1| Dirichlet L-function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 21, | "id": 21, | ||
"pid": 71, | "pid": 71, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 22 | |- | ||
| 22 | |||
| <syntaxhighlight lang="tex" inline >\operatorname{Bi}'(z)\sim \frac{z^{\frac{1}{4}}e^{\frac{2}{3}z^{\frac{3}{2}}}}{\sqrt\pi\,}\left[ \sum_{n=0}^{\infty}\frac{1+6n}{1-6n} \dfrac{ \Gamma(n+\frac{5}{6})\Gamma(n+\frac{1}{6})\left(\frac{3}{4}\right)^n}{2\pi n! z^{3n/2}} \right]</syntaxhighlight> | |||
| [[Airy function#math.72.15| Airy function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 22, | "id": 22, | ||
"pid": 72, | "pid": 72, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 23 | |- | ||
| 23 | |||
| <syntaxhighlight lang="tex" inline >F'(y)=1-2yF(y)</syntaxhighlight> | |||
| [[Dawson function#math.73.41| Dawson function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 23, | "id": 23, | ||
"pid": 73, | "pid": 73, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 24 | |- | ||
| 24 | |||
| <syntaxhighlight lang="tex" inline >s\not =1</syntaxhighlight> | |||
| [[Hurwitz zeta function#math.74.0| Hurwitz zeta function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 24, | "id": 24, | ||
"pid": 74, | "pid": 74, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 25 | |- | ||
| 25 | |||
| <syntaxhighlight lang="tex" inline >q = e^{i\pi\tau}</syntaxhighlight> | |||
| [[Theta function#math.75.6| Theta function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 25, | "id": 25, | ||
"pid": 75, | "pid": 75, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 26 | |- | ||
| 26 | |||
| <syntaxhighlight lang="tex" inline >\frac{\mathrm{d}}{\mathrm{d}z} \operatorname{dn}(z) = - k^2 \operatorname{sn}(z) \operatorname{cn}(z)</syntaxhighlight> | |||
| [[Jacobi elliptic functions#math.76.155| Jacobi elliptic functions]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 26, | "id": 26, | ||
"pid": 76, | "pid": 76, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 27 | |- | ||
| 27 | |||
| <syntaxhighlight lang="tex" inline >\int_{-\infty}^\infty \frac {\gamma\left(\frac s 2, z^2 \pi \right)} {(z^2 \pi)^\frac s 2} e^{-2 \pi i k z} \mathrm d z = \frac {\Gamma\left(\frac {1-s} 2, k^2 \pi \right)} {(k^2 \pi)^\frac {1-s} 2}</syntaxhighlight> | |||
| [[Incomplete gamma function#math.77.118| Incomplete gamma function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 27, | "id": 27, | ||
"pid": 77, | "pid": 77, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 28 | |- | ||
| 28 | |||
| <syntaxhighlight lang="tex" inline >_{1}(z) =</syntaxhighlight> | |||
| [[Polylogarithm#math.78.0| Polylogarithm]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 28, | "id": 28, | ||
"pid": 78, | "pid": 78, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 29 | |- | ||
| 29 | |||
| <syntaxhighlight lang="tex" inline >\int_{-\infty}^\infty \operatorname{sinc}(t) \, e^{-i 2 \pi f t}\,dt = \operatorname{rect}(f)</syntaxhighlight> | |||
| [[Sinc function#math.79.11| Sinc function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 29, | "id": 29, | ||
"pid": 79, | "pid": 79, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 30 | |- | ||
| 30 | |||
| <syntaxhighlight lang="tex" inline >N=1</syntaxhighlight> | |||
| [[Exponential integral#math.80.26| Exponential integral]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 30, | "id": 30, | ||
"pid": 80, | "pid": 80, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 31 | |- | ||
| 31 | |||
| <syntaxhighlight lang="tex" inline >\sum_{n=0}^\infty \frac{n!\,\Gamma\left(\alpha + 1\right)}{\Gamma\left(n+\alpha+1\right)}L_n^{(\alpha)}(x)L_n^{(\alpha)}(y)t^n=\frac{1}{(1-t)^{\alpha + 1}}e^{-(x+y)t/(1-t)}\,_0F_1\left(;\alpha + 1;\frac{xyt}{(1-t)^2}\right)</syntaxhighlight> | |||
| [[Laguerre polynomials#math.81.84| Laguerre polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 31, | "id": 31, | ||
"pid": 81, | "pid": 81, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 32 | |- | ||
| 32 | |||
| <syntaxhighlight lang="tex" inline >c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!}</syntaxhighlight> | |||
| [[Associated Legendre polynomials#math.82.8| Associated Legendre polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 32, | "id": 32, | ||
"pid": 82, | "pid": 82, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 33 | |- | ||
| 33 | |||
| <syntaxhighlight lang="tex" inline >\mathrm{Gi}(x) = \frac{1}{\pi} \int_0^\infty \sin\left(\frac{t^3}{3} + xt\right)\, dt</syntaxhighlight> | |||
| [[Scorer's function#math.83.3| Scorer's function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 33, | "id": 33, | ||
"pid": 83, | "pid": 83, | ||
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] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 34 | |- | ||
| 34 | |||
| <syntaxhighlight lang="tex" inline >\frac{\partial^2}{\partial x^2} V(x;\sigma,\gamma)= \frac{x^2-\gamma^2-\sigma^2}{\sigma^4} \frac{\operatorname{Re}[w(z)]}{\sigma\sqrt{2 \pi}}-\frac{2 x \gamma}{\sigma^4} \frac{\operatorname{Im}[w(z)]}{\sigma\sqrt{2 \pi}}+\frac{\gamma}{\sigma^4}\frac{1}{\pi}</syntaxhighlight> | |||
| [[Voigt profile#math.84.31| Voigt profile]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 34, | "id": 34, | ||
"pid": 84, | "pid": 84, | ||
| Line 2,544: | Line 2,680: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 35 | |- | ||
| 35 | |||
| <syntaxhighlight lang="tex" inline >\Phi(z,s,a) = \frac{1}{1-z} \frac{1}{a^{s}} + \sum_{n=1}^{N-1} \frac{(-1)^{n} \mathrm{Li}_{-n}(z)}{n!} \frac{(s)_{n}}{a^{n+s}} +O(a^{-N-s})</syntaxhighlight> | |||
| [[Lerch zeta function#math.85.57| Lerch zeta function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 85, | "pid": 85, | ||
| Line 2,605: | Line 2,745: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 36 | |- | ||
| 36 | |||
| <syntaxhighlight lang="tex" inline >M(1,2,z)=(e^z-1)/z,\ \ M(1,3,z)=2!(e^z-1-z)/z^2</syntaxhighlight> | |||
| [[Confluent hypergeometric function#math.86.44| Confluent hypergeometric function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 86, | "pid": 86, | ||
| Line 2,664: | Line 2,808: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 37 | |- | ||
| 37 | |||
| <syntaxhighlight lang="tex" inline >\sigma = \pm 1</syntaxhighlight> | |||
| [[Mathieu function#math.87.54| Mathieu function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 87, | "pid": 87, | ||
| Line 2,729: | Line 2,877: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 38 | |- | ||
| 38 | |||
| <syntaxhighlight lang="tex" inline >\frac{d^2f}{dz^2} + \left(\tilde{a}z^2+\tilde{b}z+\tilde{c}\right)f=0</syntaxhighlight> | |||
| [[Parabolic cylinder function#math.88.0| Parabolic cylinder function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 88, | "pid": 88, | ||
| Line 2,779: | Line 2,931: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 39 | |- | ||
| 39 | |||
| <syntaxhighlight lang="tex" inline >c=\infty</syntaxhighlight> | |||
| [[Painlevé transcendents#math.89.23| Painlevé transcendents]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 89, | "pid": 89, | ||
| Line 2,832: | Line 2,988: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 40 | |- | ||
| 40 | |||
| <syntaxhighlight lang="tex" inline >c = a + 1</syntaxhighlight> | |||
| [[Hypergeometric function#math.90.7| Hypergeometric function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 90, | "pid": 90, | ||
| Line 2,874: | Line 3,034: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 41 | |- | ||
| 41 | |||
| <syntaxhighlight lang="tex" inline >\frac{1}{\Gamma(z)}= z e^{\gamma z} \prod_{k=1}^\infty \left\{ \left(1+\frac{z}{k}\right)e^{-z/k} \right\}</syntaxhighlight> | |||
| [[Barnes G-function#math.91.47| Barnes G-function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 91, | "pid": 91, | ||
| Line 2,954: | Line 3,118: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 42 | |- | ||
| 42 | |||
| <syntaxhighlight lang="tex" inline >192/24 = 8 = 2 \times 4</syntaxhighlight> | |||
| [[Heun function#math.92.1| Heun function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 92, | "pid": 92, | ||
| Line 3,182: | Line 3,350: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 43 | |- | ||
| 43 | |||
| <syntaxhighlight lang="tex" inline >=2</syntaxhighlight> | |||
| [[Gegenbauer polynomials#math.93.0| Gegenbauer polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 93, | "pid": 93, | ||
| Line 3,223: | Line 3,395: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 44 | |- | ||
| 44 | |||
| <syntaxhighlight lang="tex" inline >\lim_{q\to 1}\;_{j}\phi_k \left[\begin{matrix} q^{a_1} & q^{a_2} & \ldots & q^{a_j} \\ q^{b_1} & q^{b_2} & \ldots & q^{b_k} \end{matrix} ; q,(q-1)^{1+k-j} z \right]=\;_{j}F_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_j \\ b_1 & b_2 & \ldots & b_k \end{matrix} ;z \right]</syntaxhighlight> | |||
| [[Basic hypergeometric series#math.94.4| Basic hypergeometric series]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 94, | "pid": 94, | ||
| Line 3,269: | Line 3,445: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 45 | |- | ||
| 45 | |||
| <syntaxhighlight lang="tex" inline >\frac{d^2w}{dz^2}+\left(-\frac{1}{4}+\frac{\kappa}{z}+\frac{1/4-\mu^2}{z^2}\right)w=0</syntaxhighlight> | |||
| [[Whittaker function#math.95.0| Whittaker function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 95, | "pid": 95, | ||
| Line 3,316: | Line 3,496: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 46 | |- | ||
| 46 | |||
| <syntaxhighlight lang="tex" inline >e_1=\tfrac12,\qquad e_2=0,\qquad e_3=-\tfrac12</syntaxhighlight> | |||
| [[Lemniscatic elliptic function#math.96.1| Lemniscatic elliptic function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 96, | "pid": 96, | ||
| Line 3,359: | Line 3,543: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 47 | |- | ||
| 47 | |||
| <syntaxhighlight lang="tex" inline >\gamma> 0,n-p=m-q> 0</syntaxhighlight> | |||
| [[Meijer G-function#math.98.53| Meijer G-function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 98, | "pid": 98, | ||
| Line 3,405: | Line 3,593: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 48 | |- | ||
| 48 | |||
| <syntaxhighlight lang="tex" inline >\begin{pmatrix} j \\ m \quad m'\end{pmatrix}:= \sqrt{2 j + 1}\begin{pmatrix} j & 0 & j \\ m & 0 & m'\end{pmatrix}= (-1)^{j - m'} \delta_{m, -m'}</syntaxhighlight> | |||
| [[3-j symbol#math.99.30| 3-j symbol]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 99, | "pid": 99, | ||
| Line 3,452: | Line 3,644: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 49 | |- | ||
| 49 | |||
| <syntaxhighlight lang="tex" inline >\begin{Bmatrix} i & j & \ell\\ k & m & n \end{Bmatrix}= (\Phi_{i,j}^{k,m})_{\ell,n}</syntaxhighlight> | |||
| [[6-j symbol#math.100.14| 6-j symbol]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 100, | "pid": 100, | ||
| Line 3,506: | Line 3,702: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 50 | |- | ||
| 50 | |||
| <syntaxhighlight lang="tex" inline >\sum_{j_7 j_8} (2j_7+1)(2j_8+1) \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} \begin{Bmatrix} j_1 & j_2 & j_3'\\ j_4 & j_5 & j_6'\\ j_7 & j_8 & j_9 \end{Bmatrix} = \frac{\delta_{j_3j_3'}\delta_{j_6j_6'} \begin{Bmatrix} j_1 & j_2 & j_3 \end{Bmatrix} \begin{Bmatrix} j_4 & j_5 & j_6\end{Bmatrix} \begin{Bmatrix} j_3 & j_6 & j_9 \end{Bmatrix}} {(2j_3+1)(2j_6+1)}</syntaxhighlight> | |||
| [[9-j symbol#math.101.32| 9-j symbol]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 101, | "pid": 101, | ||
| Line 3,553: | Line 3,753: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 51 | |- | ||
| 51 | |||
| <syntaxhighlight lang="tex" inline >\mathcal{K}_k(x; n,q) = \sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \binom {n-j}{k-j} \binom{x}{j}</syntaxhighlight> | |||
| [[Kravchuk polynomials#math.102.5| Kravchuk polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 102, | "pid": 102, | ||
| Line 3,596: | Line 3,800: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 52 | |- | ||
| 52 | |||
| <syntaxhighlight lang="tex" inline >g_1(x) = \sum_{k \geq 1} \frac{\sin(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2</syntaxhighlight> | |||
| [[Kelvin functions#math.103.8| Kelvin functions]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 103, | "pid": 103, | ||
| Line 3,651: | Line 3,859: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 53 | |- | ||
| 53 | |||
| <syntaxhighlight lang="tex" inline >S_{\mu,\nu}(z) = s_{\mu,\nu}(z) + 2^{\mu-1} \Gamma\left(\frac{\mu + \nu + 1}{2}\right) \Gamma\left(\frac{\mu - \nu + 1}{2}\right)\left(\sin \left[(\mu - \nu)\frac{\pi}{2}\right] J_\nu(z) - \cos \left[(\mu - \nu)\frac{\pi}{2}\right] Y_\nu(z)\right)</syntaxhighlight> | |||
| [[Lommel function#math.104.2| Lommel function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 104, | "pid": 104, | ||
| Line 3,703: | Line 3,915: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 54 | |- | ||
| 54 | |||
| <syntaxhighlight lang="tex" inline >\mathbf{H}_{\alpha}(z) = \frac{z^{\alpha+1}}{2^{\alpha}\sqrt{\pi} \Gamma \left (\alpha+\tfrac{3}{2} \right )} {}_1F_2 \left (1,\tfrac{3}{2}, \alpha+\tfrac{3}{2},-\tfrac{z^2}{4} \right )</syntaxhighlight> | |||
| [[Struve function#math.105.18| Struve function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 105, | "pid": 105, | ||
| Line 3,764: | Line 3,980: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 55 | |- | ||
| 55 | |||
| <syntaxhighlight lang="tex" inline >f(t+p) = f(t)</syntaxhighlight> | |||
| [[Hill differential equation#math.106.7| Hill differential equation]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 106, | "pid": 106, | ||
| Line 3,814: | Line 4,034: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 56 | |- | ||
| 56 | |||
| <syntaxhighlight lang="tex" inline >\mathbf{J}_\nu(z)=\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}+\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}</syntaxhighlight> | |||
| [[Anger function#math.108.3| Anger function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 108, | "pid": 108, | ||
| Line 3,865: | Line 4,089: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 57 | |- | ||
| 57 | |||
| <syntaxhighlight lang="tex" inline >(\operatorname{Ec})^'_{2K} = (\operatorname{Ec})^'_0 = 0, \;\; (\operatorname{Es})^'_{2K} = (\operatorname{Es})^'_0 = 0</syntaxhighlight> | |||
| [[Lamé function#math.109.27| Lamé function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 109, | "pid": 109, | ||
| Line 3,904: | Line 4,132: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 58 | |- | ||
| 58 | |||
| <syntaxhighlight lang="tex" inline >\int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)</syntaxhighlight> | |||
| [[Gauss–Hermite quadrature#math.110.1| Gauss–Hermite quadrature]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 110, | "pid": 110, | ||
| Line 3,957: | Line 4,189: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 59 | |- | ||
| 59 | |||
| <syntaxhighlight lang="tex" inline >p_n(x;a,b,c,d|q) =(ab,ac,ad;q)_na^{-n}\;_{4}\phi_3 \left[\begin{matrix} q^{-n}&abcdq^{n-1}&ae^{i\theta}&ae^{-i\theta} \\ ab&ac&ad \end{matrix} ; q,q \right]</syntaxhighlight> | |||
| [[Askey–Wilson polynomials#math.111.0| Askey–Wilson polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 111, | "pid": 111, | ||
| Line 4,005: | Line 4,241: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 60 | |- | ||
| 60 | |||
| <syntaxhighlight lang="tex" inline >Q_n(x;\alpha,\beta,N)= {}_3F_2(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1).</syntaxhighlight> | |||
| [[Hahn polynomials#math.112.0| Hahn polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 60, | "id": 60, | ||
"pid": 112, | "pid": 112, | ||
| Line 4,052: | Line 4,292: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 61 | |- | ||
| 61 | |||
| <syntaxhighlight lang="tex" inline >\sum_{x=0}^\infty \frac{\mu^x}{x!} C_n(x; \mu)C_m(x; \mu)=\mu^{-n} e^\mu n! \delta_{nm}, \quad \mu>0</syntaxhighlight> | |||
| [[Charlier polynomials#math.113.2| Charlier polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 113, | "pid": 113, | ||
| Line 4,092: | Line 4,336: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 62 | |- | ||
| 62 | |||
| <syntaxhighlight lang="tex" inline >p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q) = {}_4\phi_3\left[\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&q^{x+1}cd\\aq&bdq&cq\\ \end{matrix};q;q\right]</syntaxhighlight> | |||
| [[Q-Racah polynomials#math.114.0| Q-Racah polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 114, | "pid": 114, | ||
| Line 4,124: | Line 4,372: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 63 | |- | ||
| 63 | |||
| <syntaxhighlight lang="tex" inline >\displaystyle c_n(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a)</syntaxhighlight> | |||
| [[Q-Charlier polynomials#math.115.0| Q-Charlier polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 115, | "pid": 115, | ||
| Line 4,160: | Line 4,412: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 64 | |- | ||
| 64 | |||
| <syntaxhighlight lang="tex" inline >M_n(x,\beta,\gamma) = \sum_{k=0}^n (-1)^k{n \choose k}{x\choose k}k!(x+\beta)_{n-k}\gamma^{-k}</syntaxhighlight> | |||
| [[Meixner polynomials#math.116.0| Meixner polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 64, | "id": 64, | ||
"pid": 116, | "pid": 116, | ||
| Line 4,204: | Line 4,460: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 65 | |- | ||
| 65 | |||
| <syntaxhighlight lang="tex" inline >x(1-x) \frac {\partial^2F_1(x,y)} {\partial x^2} + y(1-x) \frac {\partial^2F_1(x,y)} {\partial x \partial y} + [c - (a+b_1+1) x] \frac {\partial F_1(x,y)} {\partial x} - b_1 y \frac {\partial F_1(x,y)} {\partial y} - a b_1 F_1(x,y) = 0</syntaxhighlight> | |||
| [[Appell series#math.117.19| Appell series]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 117, | "pid": 117, | ||
| Line 4,262: | Line 4,522: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 66 | |- | ||
| 66 | |||
| <syntaxhighlight lang="tex" inline >\Theta_\Lambda(\tau) = \sum_{x\in\Lambda}e^{i\pi\tau\|x\|^2}\qquad\mathrm{Im}\,\tau > 0</syntaxhighlight> | |||
| [[Theta function of a lattice#math.118.0| Theta function of a lattice]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
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"pid": 118, | "pid": 118, | ||
| Line 4,308: | Line 4,572: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 67 | |- | ||
| 67 | |||
| <syntaxhighlight lang="tex" inline >\frac{d^2 S}{dz^2}+\left(\sum _{j=1}^N \frac{\gamma _j}{z - a_j} \right) \frac{dS}{dz} + \frac{V(z)}{\prod _{j=1}^N (z - a_j)}S = 0</syntaxhighlight> | |||
| [[Heine–Stieltjes polynomials#math.119.0| Heine–Stieltjes polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 67, | "id": 67, | ||
"pid": 119, | "pid": 119, | ||
| Line 4,376: | Line 4,644: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 68 | |- | ||
| 68 | |||
| <syntaxhighlight lang="tex" inline >w(x) = \frac{k}{\sqrt{\pi}} x^{-1/2} \exp(-k^2\log^2 x)</syntaxhighlight> | |||
| [[Stieltjes–Wigert polynomials#math.120.0| Stieltjes–Wigert polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 68, | "id": 68, | ||
"pid": 120, | "pid": 120, | ||
| Line 4,449: | Line 4,721: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 69 | |- | ||
| 69 | |||
| <syntaxhighlight lang="tex" inline >y^2=x(x-1)(x-\lambda)</syntaxhighlight> | |||
| [[Modular lambda function#math.121.23| Modular lambda function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 69, | "id": 69, | ||
"pid": 121, | "pid": 121, | ||
| Line 4,498: | Line 4,774: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 70 | |- | ||
| 70 | |||
| <syntaxhighlight lang="tex" inline >P_1^{(\lambda)}(x;\phi)=2(\lambda\cos\phi + x\sin\phi)</syntaxhighlight> | |||
| [[Meixner–Pollaczek polynomials#math.122.3| Meixner–Pollaczek polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 70, | "id": 70, | ||
"pid": 122, | "pid": 122, | ||
| Line 4,532: | Line 4,812: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 71 | |- | ||
| 71 | |||
| <syntaxhighlight lang="tex" inline >P_n^{(\alpha,\beta)}(z)=\frac{(\alpha+1)_n}{n!}\,{}_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\tfrac{1}{2}(1-z)\right)</syntaxhighlight> | |||
| [[Jacobi polynomials#math.123.0| Jacobi polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 71, | "id": 71, | ||
"pid": 123, | "pid": 123, | ||
| Line 4,587: | Line 4,871: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 72 | |- | ||
| 72 | |||
| <syntaxhighlight lang="tex" inline >S_n(x^2;a,b,c)= {}_3F_2(-n,a+ix,a-ix;a+b,a+c;1).</syntaxhighlight> | |||
| [[Continuous dual Hahn polynomials#math.124.0| Continuous dual Hahn polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 72, | "id": 72, | ||
"pid": 124, | "pid": 124, | ||
| Line 4,633: | Line 4,921: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 73 | |- | ||
| 73 | |||
| <syntaxhighlight lang="tex" inline >P_n^{(\alpha,\beta)}=\lim_{t\to\infty}t^{-n}p_n\left(\tfrac12xt; \tfrac12(\alpha+1-it), \tfrac12(\beta+1+it), \tfrac12(\alpha+1+it), \tfrac12(\beta+1-it)\right)</syntaxhighlight> | |||
| [[Continuous Hahn polynomials#math.125.15| Continuous Hahn polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 73, | "id": 73, | ||
"pid": 125, | "pid": 125, | ||
| Line 4,681: | Line 4,973: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 74 | |- | ||
| 74 | |||
| <syntaxhighlight lang="tex" inline >\sum^{b-1}_{s=a}w_n^{(c)}(s,a,b)w_m^{(c)}(s,a,b)\rho(s)[\Delta x(s-\frac{1}{2}) ]=\delta_{nm}d_n^2</syntaxhighlight> | |||
| [[Dual Hahn polynomials#math.126.7| Dual Hahn polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 74, | "id": 74, | ||
"pid": 126, | "pid": 126, | ||
| Line 4,715: | Line 5,011: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 75 | |- | ||
| 75 | |||
| <syntaxhighlight lang="tex" inline >p_n(x;a,b,c|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_n*_4\Phi_3(q^{-n},abcdq^{n-1},ae^{i{(t+2u)}},ae^{-it};abe^{2iu},ac,ad;q;q)</syntaxhighlight> | |||
| [[Continuous q-Hahn polynomials#math.127.0| Continuous q-Hahn polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 75, | "id": 75, | ||
"pid": 127, | "pid": 127, | ||
| Line 4,769: | Line 5,069: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 76 | |- | ||
| 76 | |||
| <syntaxhighlight lang="tex" inline >p_n(x;a,b,c\mid q)=\frac{(ab,ac;q)_n}{a^n}\cdot {_3\Phi_2}(q^-n,ae^{i\theta},ae^{-i\theta}; ab, ac \mid q;q)</syntaxhighlight> | |||
| [[Continuous dual q-Hahn polynomials#math.128.0| Continuous dual q-Hahn polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 76, | "id": 76, | ||
"pid": 128, | "pid": 128, | ||
| Line 4,819: | Line 5,123: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 77 | |- | ||
| 77 | |||
| <syntaxhighlight lang="tex" inline >Q_n(x;a,b,N;q)=\;_{3}\phi_2\left[\begin{matrix} q^-n & abq^n+1 & x \\ aq & q^-N \end{matrix} ; q,q \right]</syntaxhighlight> | |||
| [[Q-Hahn polynomials#math.129.0| Q-Hahn polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 77, | "id": 77, | ||
"pid": 129, | "pid": 129, | ||
| Line 4,867: | Line 5,175: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 78 | |- | ||
| 78 | |||
| <syntaxhighlight lang="tex" inline >x=</syntaxhighlight> | |||
| [[Al-Salam–Chihara polynomials#math.131.0| Al-Salam–Chihara polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 78, | "id": 78, | ||
"pid": 131, | "pid": 131, | ||
| Line 4,910: | Line 5,222: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 79 | |- | ||
| 79 | |||
| <syntaxhighlight lang="tex" inline >\Phi_n^*(z)=z^n\overline{\Phi_n(1/\overline{z})}</syntaxhighlight> | |||
| [[Orthogonal polynomials on the unit circle#math.132.7| Orthogonal polynomials on the unit circle]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 79, | "id": 79, | ||
"pid": 132, | "pid": 132, | ||
| Line 4,950: | Line 5,266: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 80 | |- | ||
| 80 | |||
| <syntaxhighlight lang="tex" inline >P_n(x) = c_n \, \det \begin{bmatrix}m_0 & m_1 & m_2 &\cdots & m_n \\m_1 & m_2 & m_3 &\cdots & m_{n+1} \\&&\vdots&& \vdots \\m_{n-1} &m_n& m_{n+1} &\cdots &m_{2n-1}\\1 & x & x^2 & \cdots & x^n\end{bmatrix}</syntaxhighlight> | |||
| [[Orthogonal polynomials#math.133.8| Orthogonal polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 80, | "id": 80, | ||
"pid": 133, | "pid": 133, | ||
| Line 4,998: | Line 5,318: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 81 | |- | ||
| 81 | |||
| <syntaxhighlight lang="tex" inline >\displaystyle p_n(x;a,b;q) = {}_2\phi_1(q^{-n},abq^{n+1};aq;q,xq)</syntaxhighlight> | |||
| [[Little q-Jacobi polynomials#math.134.0| Little q-Jacobi polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 81, | "id": 81, | ||
"pid": 134, | "pid": 134, | ||
| Line 5,049: | Line 5,373: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 82 | |- | ||
| 82 | |||
| <syntaxhighlight lang="tex" inline >\displaystyle P_n(x;a,b,c;q)={}_3\phi_2(q^{-n},abq^{n+1},x;aq,cq;q,q)</syntaxhighlight> | |||
| [[Big q-Jacobi polynomials#math.135.0| Big q-Jacobi polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 82, | "id": 82, | ||
"pid": 135, | "pid": 135, | ||
| Line 5,096: | Line 5,424: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 83 | |- | ||
| 83 | |||
| <syntaxhighlight lang="tex" inline >P_n(x;a,b;q)=\frac{1}{(b^{-1}*q^{-n};q,n)}*_2\Phi_1(q^{-n},aqx^{-1};aq|q;\frac{x}{b})</syntaxhighlight> | |||
| [[Big q-Laguerre polynomials#math.137.0| Big q-Laguerre polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 83, | "id": 83, | ||
"pid": 137, | "pid": 137, | ||
| Line 5,150: | Line 5,482: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 84 | |- | ||
| 84 | |||
| <syntaxhighlight lang="tex" inline >K_n(\lambda(x);c,N|q)=_3\Phi_2(q^{-n},q^{-x},cq^{x-N};q^{-N},0|q;q)</syntaxhighlight> | |||
| [[Dual q-Krawtchouk polynomials#math.138.0| Dual q-Krawtchouk polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 84, | "id": 84, | ||
"pid": 138, | "pid": 138, | ||
| Line 5,196: | Line 5,532: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 85 | |- | ||
| 85 | |||
| <syntaxhighlight lang="tex" inline >P_{n}^{(\alpha)}(x|q)=\frac{(q^\alpha+1;q)_{n}}{(q;q)_{n}}</syntaxhighlight> | |||
| [[Continuous q-Laguerre polynomials#math.139.0| Continuous q-Laguerre polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 85, | "id": 85, | ||
"pid": 139, | "pid": 139, | ||
| Line 5,242: | Line 5,582: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 86 | |- | ||
| 86 | |||
| <syntaxhighlight lang="tex" inline >\displaystyle p_n(x;a|q) = {}_2\phi_1(q^{-n},0;aq;q,qx) = \frac{1}{(a^{-1}q^{-n};q)_n}{}_2\phi_0(q^{-n},x^{-1};;q,x/a)</syntaxhighlight> | |||
| [[Little q-Laguerre polynomials#math.142.0| Little q-Laguerre polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 86, | "id": 86, | ||
"pid": 142, | "pid": 142, | ||
| Line 5,295: | Line 5,639: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 87 | |- | ||
| 87 | |||
| <syntaxhighlight lang="tex" inline >y_{n}(x;a;q)=\;_{2}\phi_1 \left(\begin{matrix} q^{-N} & -aq^{n} \\ 0 \end{matrix} ; q,qx \right)</syntaxhighlight> | |||
| [[Q-Bessel polynomials#math.143.0| Q-Bessel polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 87, | "id": 87, | ||
"pid": 143, | "pid": 143, | ||
| Line 5,341: | Line 5,689: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 88 | |- | ||
| 88 | |||
| <syntaxhighlight lang="tex" inline >h_n(ix;q^{-1}) = i^n\hat h_n(x;q)</syntaxhighlight> | |||
| [[Discrete q-Hermite polynomials#math.144.2| Discrete q-Hermite polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 88, | "id": 88, | ||
"pid": 144, | "pid": 144, | ||
| Line 5,393: | Line 5,745: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 89 | |- | ||
| 89 | |||
| <syntaxhighlight lang="tex" inline >P_{n}(x;a\mid q) = a^{-n} e^{in\phi} \frac{a^2;q_n}{(q;q)_n} {_3}\Phi_2(q^-n, ae^{i(\theta+2\phi)}, ae^{-i\theta}; a^2, 0 \mid q; q)</syntaxhighlight> | |||
| [[Q-Meixner–Pollaczek polynomials#math.145.0| Q-Meixner–Pollaczek polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 89, | "id": 89, | ||
"pid": 145, | "pid": 145, | ||
| Line 5,445: | Line 5,801: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 90 | |- | ||
| 90 | |||
| <syntaxhighlight lang="tex" inline >\displaystyle L_n^{(\alpha)}(x;q) = \frac{(q^{\alpha+1};q)_n}{(q;q)_n} {}_1\phi_1(q^{-n};q^{\alpha+1};q,-q^{n+\alpha+1}x)</syntaxhighlight> | |||
| [[Q-Laguerre polynomials#math.149.0| Q-Laguerre polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 90, | "id": 90, | ||
"pid": 149, | "pid": 149, | ||
| Line 5,500: | Line 5,860: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 91 | |- | ||
| 91 | |||
| <syntaxhighlight lang="tex" inline >\sum_{n=0}^\infty H_n(x \mid q) \frac{t^n}{(q;q)_n} = \frac{1}{\left( t e^{i \theta},t e^{-i \theta};q \right)_\infty}</syntaxhighlight> | |||
| [[Continuous q-Hermite polynomials#math.150.3| Continuous q-Hermite polynomials]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 91, | "id": 91, | ||
"pid": 150, | "pid": 150, | ||
| Line 5,538: | Line 5,902: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 92 | |- | ||
| 92 | |||
| <syntaxhighlight lang="tex" inline >w^{\prime\prime}+\xi\sin(2z)w^{\prime}+(\eta-p\xi\cos(2z))w=0. </syntaxhighlight> | |||
| [[Ince equation#math.151.0| Ince equation]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 92, | "id": 92, | ||
"pid": 151, | "pid": 151, | ||
| Line 5,600: | Line 5,968: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 93 | |- | ||
| 93 | |||
| <syntaxhighlight lang="tex" inline >Q_v^\mu(x)= \cos(\mu\pi)\left(\frac{1+x}{1-x}\right)^{\mu/2}\frac{F(v+1,-v;1-\mu;1/2-2/x)} {\Gamma(1-\mu ) }</syntaxhighlight> | |||
| [[Ferrers function#math.152.1| Ferrers function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 93, | "id": 93, | ||
"pid": 152, | "pid": 152, | ||
| Line 5,651: | Line 6,023: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 94 | |- | ||
| 94 | |||
| <syntaxhighlight lang="tex" inline >H_{-v}^{(1)}(z,w)=e^{v\pi i}H_v^{(1)}(z,w)</syntaxhighlight> | |||
| [[Incomplete Bessel functions#math.153.27| Incomplete Bessel functions]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 94, | "id": 94, | ||
"pid": 153, | "pid": 153, | ||
| Line 5,687: | Line 6,063: | ||
] | ] | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div>|- | </div></div> | ||
| 95 | |- | ||
| 95 | |||
| <syntaxhighlight lang="tex" inline >K_v(x,y)=\int_1^\infty\frac{e^{-xt-\frac{y}{t}}}{t^{v+1}}dt</syntaxhighlight> | |||
| [[Incomplete Bessel K function/generalized incomplete gamma function#math.154.0| Incomplete Bessel K function/generalized incomplete gamma function]] | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | |||
"id": 95, | "id": 95, | ||
"pid": 154, | "pid": 154, | ||
Revision as of 12:14, 6 February 2021
| # | Formula | Title | Evaluation data |
|---|---|---|---|
| 1 | \begin{align}J_{-(m+\frac{1}{2})}(x) &= (-1)^{m+1} Y_{m+\frac{1}{2}}(x), \\Y_{-(m+\frac{1}{2})}(x) &= (-1)^m J_{m+\frac{1}{2}}(x).\end{align}
|
Bessel function | Full data:
{
"id": 1,
"pid": 51,
"eid": "math.51.18",
"title": "Bessel function",
"formulae": [
{
"id": "FORMULA_0f521573a47e7fd187dafed615b0ecce",
"formula": "\\begin{align}J_{-(m+\\frac{1}{2})}(x) &= (-1)^{m+1} Y_{m+\\frac{1}{2}}(x), \\\\Y_{-(m+\\frac{1}{2})}(x) &= (-1)^m J_{m+\\frac{1}{2}}(x).\\end{align}",
"semanticFormula": "\\begin{align}\\BesselJ{- (m + \\frac{1}{2})}@{x} &= (- 1)^{m+1} \\BesselY{m+\\frac{1}{2}}@{x} , \\\\ \\BesselY{- (m + \\frac{1}{2})}@{x} &= (-1)^m \\BesselJ{m+\\frac{1}{2}}@{x} .\\end{align}",
"confidence": 0.8803349492974287,
"translations": {
"Mathematica": {
"translation": "BesselJ[- (m +Divide[1,2]), x] == (- 1)^(m + 1)* BesselY[m +Divide[1,2], x]\nBesselY[- (m +Divide[1,2]), x] == (- 1)^(m)* BesselJ[m +Divide[1,2], x]",
"translationInformation": {
"subEquations": [
"BesselJ[- (m +Divide[1,2]), x] = (- 1)^(m + 1)* BesselY[m +Divide[1,2], x]",
"BesselY[- (m +Divide[1,2]), x] = (- 1)^(m)* BesselJ[m +Divide[1,2], x]"
],
"freeVariables": [
"m",
"x"
],
"constraints": [],
"tokenTranslations": {
"\\pgcd": "Greatest common divisor; Example: \\pgcd{m,n}\nWill be translated to: GCD[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/27.1#p2.t1.r3\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/GCD.html",
"\\BesselY": "Bessel function second kind; Example: \\BesselY{v}@{z}\nWill be translated to: BesselY[$0, $1]\nBranch Cuts: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/10.2#E3\nMathematica: https:\/\/",
"\\BesselJ": "Bessel function first kind; Example: \\BesselJ{v}@{z}\nWill be translated to: BesselJ[$0, $1]\nBranch Cuts: if v \\notin \\Integers: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/10.2#E2\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/BesselJ.html"
}
}
},
"Maple": {
"translation": "BesselJ(- (m +(1)\/(2)), x) = (- 1)^(m + 1)* BesselY(m +(1)\/(2), x); BesselY(- (m +(1)\/(2)), x) = (- 1)^(m)* BesselJ(m +(1)\/(2), x)",
"translationInformation": {
"subEquations": [
"BesselJ(- (m +(1)\/(2)), x) = (- 1)^(m + 1)* BesselY(m +(1)\/(2), x)",
"BesselY(- (m +(1)\/(2)), x) = (- 1)^(m)* BesselJ(m +(1)\/(2), x)"
],
"freeVariables": [
"m",
"x"
],
"constraints": [],
"tokenTranslations": {
"\\pgcd": "Greatest common divisor; Example: \\pgcd{m,n}\nWill be translated to: gcd($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/27.1#p2.t1.r3\nMaple: https:\/\/www.maplesoft.com\/support\/help\/Maple\/view.aspx?path=gcd",
"\\BesselY": "Bessel function second kind; Example: \\BesselY{v}@{z}\nWill be translated to: BesselY($0, $1)\nBranch Cuts: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/10.2#E3\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=Bessel",
"\\BesselJ": "Bessel function first kind; Example: \\BesselJ{v}@{z}\nWill be translated to: BesselJ($0, $1)\nBranch Cuts: if v \\notin \\Integers: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/10.2#E2\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=Bessel"
}
}
}
},
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"includes": [
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"J_{\\alpha}(x)",
"Y_{n}",
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"J_{\\alpha}",
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"(-1)^{m}",
"J_{n}",
"J_{\\alpha}(z)",
"J_{\\alpha}(k)",
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],
"isPartOf": [],
"definiens": [
{
"definition": "Bessel function first kind",
"score": 2
},
{
"definition": "Bessel function second kind",
"score": 2
},
{
"definition": "above relation",
"score": 0
},
{
"definition": "spherical Bessel",
"score": 1
},
{
"definition": "integer",
"score": 1
},
{
"definition": "nonnegative integer",
"score": 1
},
{
"definition": "relationship",
"score": 0
},
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"definition": "function",
"score": 1
},
{
"definition": "recurrence relation",
"score": 1
},
{
"definition": "Bessel",
"score": 1
},
{
"definition": "large number of other known integral",
"score": 0
},
{
"definition": "positive zero",
"score": 0
},
{
"definition": "entire function of genus",
"score": 0
},
{
"definition": "identity",
"score": 0
},
{
"definition": "orthogonality relation",
"score": 0
},
{
"definition": "Bessel function",
"score": 2
},
{
"definition": "term",
"score": 0
},
{
"definition": "real zero",
"score": 0
},
{
"definition": "similar relation",
"score": 0
},
{
"definition": "Hankel",
"score": 1
},
{
"definition": "Bessel function of the second kind",
"score": 2
},
{
"definition": "limit",
"score": 0
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{
"definition": "ordinary Bessel function",
"score": 1
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{
"definition": "case",
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{
"definition": "negative integer",
"score": 0
},
{
"definition": "integral formula",
"score": 0
},
{
"definition": "small argument",
"score": 0
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{
"definition": "average",
"score": 0
},
{
"definition": "Bessel function of the first kind",
"score": 2
},
{
"definition": "reference",
"score": 0
},
{
"definition": "series expansion",
"score": 0
},
{
"definition": "spherical Bessel function",
"score": 1
},
{
"definition": "Abel 's identity",
"score": 0
},
{
"definition": "important property of Bessel 's equation",
"score": 1
},
{
"definition": "particular Bessel",
"score": 1
},
{
"definition": "solution of Bessel 's equation",
"score": 0
},
{
"definition": "Wronskian of the solution",
"score": 0
},
{
"definition": "series",
"score": 0
},
{
"definition": "closure equation",
"score": 0
}
]
}
]
}
|
| 2 | E(e) \,=\, \int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta
|
Ellipse | Full data:
{
"id": 2,
"pid": 52,
"eid": "math.52.404",
"title": "Ellipse",
"formulae": [
{
"id": "FORMULA_d3e28ddd096754fb8e1e52aaaa4f7770",
"formula": "E(e) \\,=\\, \\int_0^{\\pi\/2}\\sqrt {1 - e^2 \\sin^2\\theta}\\ d\\theta",
"semanticFormula": "\\compellintEk@{e} = \\int_0^{\\cpi \/ 2} \\sqrt{1 - e^2 \\sin^2 \\theta} \\diff{\\theta}",
"confidence": 0.8896531556938116,
"translations": {
"Mathematica": {
"translation": "EllipticE[(e)^2] == Integrate[Sqrt[1 - (e)^(2)*(Sin[\\[Theta]])^(2)], {\\[Theta], 0, Pi\/2}, GenerateConditions->None]",
"translationInformation": {
"subEquations": [
"EllipticE[(e)^2] = Integrate[Sqrt[1 - (e)^(2)*(Sin[\\[Theta]])^(2)], {\\[Theta], 0, Pi\/2}, GenerateConditions->None]"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\cpi": "Pi was translated to: Pi",
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\compellintEk": "Legendre's complete elliptic integral of the second kind; Example: \\compellintEk@{k}\nWill be translated to: EllipticE[($0)^2]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/19.2#E8\nMathematica: https:\/\/",
"\\sin": "Sine; Example: \\sin@@{z}\nWill be translated to: Sin[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E1\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Sin.html"
}
}
},
"Maple": {
"translation": "EllipticE(e) = int(sqrt(1 - (e)^(2)*(sin(theta))^(2)), theta = 0..Pi\/2)",
"translationInformation": {
"subEquations": [
"EllipticE(e) = int(sqrt(1 - (e)^(2)*(sin(theta))^(2)), theta = 0..Pi\/2)"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
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"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\compellintEk": "Legendre's complete elliptic integral of the second kind; Example: \\compellintEk@{k}\nWill be translated to: EllipticE($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/19.2#E8\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=EllipticE",
"\\sin": "Sine; Example: \\sin@@{z}\nWill be translated to: sin($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E1\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=sin"
}
}
}
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"definition": "complete elliptic integral of the second kind",
"score": 2
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"definition": "elementary function",
"score": 1
},
{
"definition": "function",
"score": 1
},
{
"definition": "length of the semi-major axis",
"score": 2
},
{
"definition": "eccentricity",
"score": 2
},
{
"definition": "circumference",
"score": 0
},
{
"definition": "ellipse",
"score": 1
},
{
"definition": "angle",
"score": 1
},
{
"definition": "angular coordinate",
"score": 1
},
{
"definition": "center",
"score": 0
},
{
"definition": "formula",
"score": 0
},
{
"definition": "rotation angle",
"score": 0
}
]
}
]
}
|
| 3 | F(x;k) = u
|
Elliptic integral | Full data:
{
"id": 3,
"pid": 53,
"eid": "math.53.6",
"title": "Elliptic integral",
"formulae": [
{
"id": "FORMULA_04e9de23897a3b23dee1a9b7312ad99e",
"formula": "F(x;k) = u",
"semanticFormula": "\\incellintFk@{\\asin@{\\Jacobiellsnk@@{u}{k}}}{k} = u",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "EllipticF[ArcSin[JacobiSN[u, (k)^2]], (k)^2] == u",
"translationInformation": {
"subEquations": [
"EllipticF[ArcSin[JacobiSN[u, (k)^2]], (k)^2] = u"
],
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"translationInformation": {
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"E(\\varphi,k) = E\\left(\\varphi \\,|\\,k^2\\right) = E(\\sin\\varphi;k) = \\int_0^\\varphi \\sqrt{1-k^2 \\sin^2\\theta}\\,\\mathrm{d}\\theta",
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],
"definiens": [
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"definition": "inverse to the elliptic integral",
"score": 1
},
{
"definition": "Jacobian elliptic function",
"score": 2
},
{
"definition": "Legendre",
"score": 1
},
{
"definition": "normal form",
"score": 1
},
{
"definition": "trigonometric form",
"score": 1
},
{
"definition": "incomplete elliptic integral of the second kind",
"score": 0
},
{
"definition": "incomplete elliptic integral of the first kind",
"score": 2
}
]
}
]
}
|
| 4 | \frac{1}{\Gamma(z)} = \frac{i}{2\pi}\int_C (-t)^{-z}e^{-t}\,dt
|
Gamma function | Full data:
{
"id": 4,
"pid": 54,
"eid": "math.54.195",
"title": "Gamma function",
"formulae": [
{
"id": "FORMULA_19a0f00da77cc439ad679c579a295bd2",
"formula": "\\frac{1}{\\Gamma(z)} = \\frac{i}{2\\pi}\\int_C (-t)^{-z}e^{-t}\\,dt",
"semanticFormula": "\\frac{1}{\\EulerGamma@{z}} = \\frac{\\iunit}{2 \\cpi} \\int_C(- t)^{-z} \\expe^{-t} \\diff{t}",
"confidence": 0.8809245132365588,
"translations": {},
"positions": [
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"word": 9
}
],
"includes": [
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"isPartOf": [],
"definiens": [
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"definition": "related expression",
"score": 0
},
{
"definition": "integer",
"score": 0
},
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"definition": "reflection formula",
"score": 1
},
{
"definition": "end",
"score": 0
},
{
"definition": "Hankel contour",
"score": 2
},
{
"definition": "Riemann sphere",
"score": 1
},
{
"definition": "Hankel 's formula for the gamma function",
"score": 2
},
{
"definition": "gamma function",
"score": 2
},
{
"definition": "reciprocal gamma function",
"score": 2
}
]
}
]
}
|
| 5 | 2^{4} = 2 \times2 \times 2 \times 2 = 16
|
Logarithm | Full data:
{
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"pid": 55,
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"id": "FORMULA_579837194f2124b255d579031524a91c",
"formula": "2^{4} = 2 \\times2 \\times 2 \\times 2 = 16",
"semanticFormula": "2^{4} = 2 \\times2 \\times 2 \\times 2 = 16",
"confidence": 0,
"translations": {
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"translation": "(2)^(4) == 2 * 2 * 2 * 2 == 16",
"translationInformation": {
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],
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"score": 2
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]
}
]
}
|
| 6 | \psi(x) := \sum_{n=1}^\infty e^{-n^2 \pi x}
|
Riemann zeta function | Full data:
{
"id": 6,
"pid": 56,
"eid": "math.56.40",
"title": "Riemann zeta function",
"formulae": [
{
"id": "FORMULA_bd88ec58aa42c7a59bc2f4ff458a54cf",
"formula": "\\psi(x) := \\sum_{n=1}^\\infty e^{-n^2 \\pi x}",
"semanticFormula": "\\psi(x) : = \\sum_{n=1}^\\infty \\expe^{- n^2 \\cpi x}",
"confidence": 0.9073333333333333,
"translations": {
"Mathematica": {
"translation": "\\[Psi][x_] := Sum[Exp[-(n)^(2)*Pi*x], {n, 1, Infinity}]"
},
"Maple": {
"translation": "psi := (x) -> sum(exp(-(n)^(2)*Pi*x), n=1..infinity)"
}
},
"positions": [
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"score": 0
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"definition": "absolute convergence",
"score": 0
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{
"definition": "convenience",
"score": 0
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{
"definition": "inversion",
"score": 0
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{
"definition": "process",
"score": 0
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{
"definition": "stricter requirement",
"score": 0
},
{
"definition": "series",
"score": 1
},
{
"definition": "definition",
"score": 2
}
]
}
]
}
|
| 7 | \operatorname{li}(x) = \lim_{\varepsilon \to 0+} \left( \int_0^{1-\varepsilon} \frac{dt}{\ln t} + \int_{1+\varepsilon}^x \frac{dt}{\ln t} \right)
|
Logarithmic integral function | Full data:
{
"id": 7,
"pid": 57,
"eid": "math.57.2",
"title": "Logarithmic integral function",
"formulae": [
{
"id": "FORMULA_36fb8f8330168b8f8acda0dc36851474",
"formula": "\\operatorname{li}(x) = \\lim_{\\varepsilon \\to 0+} \\left( \\int_0^{1-\\varepsilon} \\frac{dt}{\\ln t} + \\int_{1+\\varepsilon}^x \\frac{dt}{\\ln t} \\right)",
"semanticFormula": "\\logint@{x} = \\lim_{\\varepsilon \\to 0+}(\\int_0^{1-\\varepsilon} \\frac{\\diff{t}}{\\ln t} + \\int_{1+\\varepsilon}^x \\frac{\\diff{t}}{\\ln t})",
"confidence": 0.8728566391293461,
"translations": {
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"translation": "LogIntegral[x] == Limit[Integrate[Divide[1,Log[t]], {t, 0, 1 - \\[CurlyEpsilon]}, GenerateConditions->None]+ Integrate[Divide[1,Log[t]], {t, 1 + \\[CurlyEpsilon], x}, GenerateConditions->None], \\[CurlyEpsilon] -> 0, Direction -> \"FromAbove\", GenerateConditions->None]",
"translationInformation": {
"subEquations": [
"LogIntegral[x] = Limit[Integrate[Divide[1,Log[t]], {t, 0, 1 - \\[CurlyEpsilon]}, GenerateConditions->None]+ Integrate[Divide[1,Log[t]], {t, 1 + \\[CurlyEpsilon], x}, GenerateConditions->None], \\[CurlyEpsilon] -> 0, Direction -> \"FromAbove\", GenerateConditions->None]"
],
"freeVariables": [
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"constraints": [],
"tokenTranslations": {
"\\logint": "Logarithmic integral; Example: \\logint@{x}\nWill be translated to: LogIntegral[$0]\nConstraints: x > 1\nMathematica uses other branch cuts: (-\\inf, 1)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/6.2#E8\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/LogIntegral.html",
"\\ln": "Natural logarithm; Example: \\ln@@{z}\nWill be translated to: Log[$0]\nConstraints: z != 0\nBranch Cuts: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.2#E2\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Log.html"
}
}
},
"Maple": {
"translation": "Li(x) = limit(int((1)\/(ln(t)), t = 0..1 - varepsilon)+ int((1)\/(ln(t)), t = 1 + varepsilon..x), varepsilon = 0, right)",
"translationInformation": {
"subEquations": [
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],
"freeVariables": [
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"constraints": [],
"tokenTranslations": {
"\\logint": "Logarithmic integral; Example: \\logint@{x}\nWill be translated to: Li($0)\nConstraints: x > 1\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/6.2#E8\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=Li",
"\\ln": "Natural logarithm; Example: \\ln@@{z}\nWill be translated to: ln($0)\nConstraints: z != 0\nBranch Cuts: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.2#E2\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=ln"
}
}
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"score": 2
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"score": 1
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{
"definition": "singularity",
"score": 1
},
{
"definition": "special function",
"score": 1
},
{
"definition": "integral representation",
"score": 1
},
{
"definition": "integral logarithm li",
"score": 2
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{
"definition": "logarithmic integral function",
"score": 2
},
{
"definition": "logarithmic integral",
"score": 2
},
{
"definition": "function li",
"score": 1
}
]
}
]
}
|
| 8 | w_{i} = \frac{1}{p'_{n}(x_{i})}\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx
|
Gaussian quadrature | Full data:
{
"id": 8,
"pid": 58,
"eid": "math.58.61",
"title": "Gaussian quadrature",
"formulae": [
{
"id": "FORMULA_8c49145544fca24efb8de07eb1275c09",
"formula": "w_{i} = \\frac{1}{p'_{n}(x_{i})}\\int_{a}^{b}\\omega(x)\\frac{p_{n}(x)}{x-x_{i}}dx",
"semanticFormula": "w_{i} = \\frac{1}{p'_{n}(x_{i})} \\int_{a}^{b} \\omega(x) \\frac{p_{n}(x)}{x-x_{i}} \\diff{x}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Subscript[w, i] = Divide[1, Subscript[p\\[Prime], n][Subscript[x, i]]]*Integrate[\\[Omega][x]*Divide[Subscript[p,n][x], x-Subscript[x,i]], {x, a, b}]"
}
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"isPartOf": [],
"definiens": [
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"definition": "yield",
"score": 0
},
{
"definition": "integral expression for the weight",
"score": 2
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"definition": "integrand",
"score": 1
},
{
"definition": "L'H\u00f4pital 's rule",
"score": 0
},
{
"definition": "limit",
"score": 0
},
{
"definition": "polynomial of degree",
"score": 0
}
]
}
]
}
|
| 9 | \begin{align}x & =ue^u, \\[5pt]\frac{dx}{du} & =(u+1)e^u.\end{align}
|
Lambert W function | Full data:
{
"id": 9,
"pid": 59,
"eid": "math.59.52",
"title": "Lambert W function",
"formulae": [
{
"id": "FORMULA_fe13643d8449f601f150fd50c0751cf2",
"formula": "\\begin{align}x & =ue^u, \\\\[5pt]\\frac{dx}{du} & =(u+1)e^u.\\end{align}",
"semanticFormula": "\\begin{align}x & =\\LambertW@{x}\\expe^{\\LambertW@{x}}, \\\\ \\deriv{x}{\\LambertW@{x}} &=(\\LambertW@{x} + 1) \\expe^{\\LambertW@{x}} .\\end{align}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "x == ProductLog[x]*(E)^(ProductLog[x])\nD[x,ProductLog[x]] = (ProductLog[x] + 1)*Exp[ProductLog[x]]",
"translationInformation": {
"subEquations": [
"x = ProductLog[x]*(E)^(ProductLog[x])",
"D[x,ProductLog[x]] = (ProductLog[x] + 1)*Exp[ProductLog[x]]"
],
"freeVariables": [
"u",
"x"
],
"constraints": [],
"tokenTranslations": {
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function."
}
}
},
"Maple": {
"translation": "x = LambertW(x)*exp(u); diff(x, [LambertW(x)$1]) = (LambertW(x) + 1)*exp(LambertW(x))",
"translationInformation": {
"subEquations": [
"x = LambertW(x)*exp(u)",
"diff(x, [LambertW(x)$1]) = (LambertW(x) + 1)*exp(LambertW(x))"
],
"freeVariables": [
"u",
"x"
],
"constraints": [],
"tokenTranslations": {
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function."
}
}
}
},
"positions": [
{
"section": 12,
"sentence": 1,
"word": 14
}
],
"includes": [
"e^{w}",
"x"
],
"isPartOf": [],
"definiens": [
{
"definition": "substitution",
"score": 2
},
{
"definition": "third identity",
"score": 0
},
{
"definition": "second identity",
"score": 1
}
]
}
]
}
|
| 10 | \frac{1}{\left| \mathbf{x}-\mathbf{x}' \right|} = \frac{1}{\sqrt{r^2+{r'}^2-2r{r'}\cos\gamma}} = \sum_{\ell=0}^\infty \frac{{r'}^\ell}{r^{\ell+1}} P_\ell(\cos \gamma)
|
Legendre polynomials | Full data:
{
"id": 10,
"pid": 60,
"eid": "math.60.57",
"title": "Legendre polynomials",
"formulae": [
{
"id": "FORMULA_8646bd0d06e9454aaa39dfc506fe54f7",
"formula": "\\frac{1}{\\left| \\mathbf{x}-\\mathbf{x}' \\right|} = \\frac{1}{\\sqrt{r^2+{r'}^2-2r{r'}\\cos\\gamma}} = \\sum_{\\ell=0}^\\infty \\frac{{r'}^\\ell}{r^{\\ell+1}} P_\\ell(\\cos \\gamma)",
"semanticFormula": "\\frac{1}{|\\mathbf{x} - \\mathbf{x} '|} = \\frac{1}{\\sqrt{r^2+{r'}^2-2r{r'}\\cos\\gamma}} = \\sum_{\\ell=0}^\\infty \\frac{{r'}^\\ell}{r^{\\ell+1}} \\LegendrepolyP{\\ell}@{\\cos \\gamma}",
"confidence": 0.808438593520797,
"translations": {
"Mathematica": "Divide[1, Abs[x - x\\[Prime]]] == Divide[1, Sqrt[r^2+(r\\[Prime])^(2)-2*r*r\\[Prime] Cos[\\[Gamma]]]] == Sum[Divide[(r\\[Prime])^(\\[ScriptL]), r^(\\[ScriptL]+1)]*LegendreP[\\[ScriptL], Cos[\\[Gamma]]], {\\[ScriptL], 0, Infinity}]"
},
"positions": [
{
"section": 6,
"sentence": 0,
"word": 21
}
],
"includes": [
"P_n(x)",
"P_n",
"P_n(\\cos\\theta)",
"P_{n}(x)",
"P_m",
"r",
"r{'}",
"\\mathbf{x}",
"\\mathbf{x}{'}",
"\\gamma",
"P"
],
"isPartOf": [],
"definiens": [
{
"definition": "expansion",
"score": 2
},
{
"definition": "Adrien-Marie Legendre as the coefficient",
"score": 0
},
{
"definition": "angle",
"score": 1
},
{
"definition": "Legendre polynomial",
"score": 2
},
{
"definition": "length of the vector",
"score": 1
},
{
"definition": "vector",
"score": 1
},
{
"definition": "polynomial",
"score": 1
}
]
}
]
}
|
| 11 | \operatorname{erf}^{(k)}(z) = \frac{2 (-1)^{k-1}}{\sqrt{\pi}} \mathit{H}_{k-1}(z) e^{-z^2} = \frac{2}{\sqrt{\pi}} \frac{d^{k-1}}{dz^{k-1}} \left(e^{-z^2}\right),\qquad k=1, 2, \dots
|
Error function | Full data:
{
"id": 11,
"pid": 61,
"eid": "math.61.27",
"title": "Error function",
"formulae": [
{
"id": "FORMULA_523ec091d0929f0fa69ae7e0d563a72b",
"formula": "\\operatorname{erf}^{(k)}(z) = \\frac{2 (-1)^{k-1}}{\\sqrt{\\pi}} \\mathit{H}_{k-1}(z) e^{-z^2} = \\frac{2}{\\sqrt{\\pi}} \\frac{d^{k-1}}{dz^{k-1}} \\left(e^{-z^2}\\right),\\qquad k=1, 2, \\dots",
"semanticFormula": "\\erf@@{(z)}^{(k)} = \\frac{2 (-1)^{k-1}}{\\sqrt{\\cpi}} \\HermitepolyH{k-1}@{z} \\expe^{-z^2} = \\frac{2}{\\sqrt{\\cpi}} \\deriv [{k-1}]{ }{z}(\\expe^{-z^2}) , \\qquad k = 1 , 2 , \\dots",
"confidence": 0.82607945540953,
"translations": {
"Mathematica": {
"translation": "D[Erf[z], {z, k}] == Divide[2*(- 1)^(k - 1),Sqrt[Pi]]*HermiteH[k - 1, z]*Exp[- (z)^(2)] == Divide[2,Sqrt[Pi]]*D[Exp[- (z)^(2)], {z, k - 1}]",
"translationInformation": {
"subEquations": [
"D[Erf[z], {z, k}] = Divide[2*(- 1)^(k - 1),Sqrt[Pi]]*HermiteH[k - 1, z]*Exp[- (z)^(2)]",
"Divide[2*(- 1)^(k - 1),Sqrt[Pi]]*HermiteH[k - 1, z]*Exp[- (z)^(2)] = Divide[2,Sqrt[Pi]]*D[Exp[- (z)^(2)], {z, k - 1}]"
],
"freeVariables": [
"k",
"z"
],
"constraints": [
"k == 1 , 2 , \\[Ellipsis]"
],
"tokenTranslations": {
"\\deriv1": "Derivative; Example: \\deriv[n]{f}{x}\nWill be translated to: D[$1, {$2, $0}]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/1.4#E4\nMathematica: https:\/\/",
"\\cpi": "Pi was translated to: Pi",
"\\HermitepolyH": "Hermite polynomial; Example: \\HermitepolyH{n}@{x}\nWill be translated to: HermiteH[$0, $1]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/18.3#T1.t1.r13\nMathematica: https:\/\/",
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\erf": "Error function; Example: \\erf@@{z}\nWill be translated to: Erf[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/7.2#E1\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Erf.html"
}
}
},
"Maple": {
"translation": "diff(erf(z), [z$k]) = (2*(- 1)^(k - 1))\/(sqrt(Pi))*HermiteH(k - 1, z)*exp(- (z)^(2)) = (2)\/(sqrt(Pi))*diff(exp(- (z)^(2)), [z$(k - 1)])",
"translationInformation": {
"subEquations": [
"diff(erf(z), [z$k]) = (2*(- 1)^(k - 1))\/(sqrt(Pi))*HermiteH(k - 1, z)*exp(- (z)^(2))",
"(2*(- 1)^(k - 1))\/(sqrt(Pi))*HermiteH(k - 1, z)*exp(- (z)^(2)) = (2)\/(sqrt(Pi))*diff(exp(- (z)^(2)), [z$(k - 1)])"
],
"freeVariables": [
"k",
"z"
],
"constraints": [
"k = 1 , 2 , .."
],
"tokenTranslations": {
"\\deriv1": "Derivative; Example: \\deriv[n]{f}{x}\nWill be translated to: diff($1, [$2$($0)])\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/1.4#E4\nMaple: https:\/\/www.maplesoft.com\/support\/help\/Maple\/view.aspx?path=diff",
"\\cpi": "Pi was translated to: Pi",
"\\HermitepolyH": "Hermite polynomial; Example: \\HermitepolyH{n}@{x}\nWill be translated to: HermiteH($0, $1)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/18.3#T1.t1.r13\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=HermiteH",
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\erf": "Error function; Example: \\erf@@{z}\nWill be translated to: erf($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/7.2#E1\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=erf"
}
}
}
},
"positions": [
{
"section": 5,
"sentence": 4,
"word": 6
}
],
"includes": [
"erf",
"e^{-t^2}",
"-1",
"z",
"z)",
"e",
"\\mathit{H}",
"z^{\\bar{n}}"
],
"isPartOf": [],
"definiens": [
{
"definition": "Higher order derivative",
"score": 2
},
{
"definition": "physicists ' Hermite polynomial",
"score": 1
},
{
"definition": "name error function",
"score": 1
},
{
"definition": "erfc",
"score": 1
},
{
"definition": "error function",
"score": 2
},
{
"definition": "erf",
"score": 1
}
]
}
]
}
|
| 12 | x_k = \cos\left(\frac{\pi(k+1/2)}{n}\right),\quad k=0,\ldots,n-1
|
Chebyshev polynomials | Full data:
{
"id": 12,
"pid": 62,
"eid": "math.62.44",
"title": "Chebyshev polynomials",
"formulae": [
{
"id": "FORMULA_d9eb68704833b0f525c4ca81a749d9ca",
"formula": "x_k = \\cos\\left(\\frac{\\pi(k+1\/2)}{n}\\right),\\quad k=0,\\ldots,n-1",
"semanticFormula": "x_k = \\cos(\\frac{\\cpi(k + 1 \/ 2)}{n}) , \\quad k = 0 , \\ldots , n - 1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Subscript[x, k] == Cos[Divide[Pi*(k + 1\/2),n]]",
"translationInformation": {
"subEquations": [
"Subscript[x, k] = Cos[Divide[Pi*(k + 1\/2),n]]"
],
"freeVariables": [
"Subscript[x, k]",
"k",
"n"
],
"constraints": [
"k == 0 , \\[Ellipsis], n - 1"
],
"tokenTranslations": {
"\\cos": "Cosine; Example: \\cos@@{z}\nWill be translated to: Cos[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E2\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Cos.html",
"\\cpi": "Pi was translated to: Pi"
}
}
},
"Maple": {
"translation": "x[k] = cos((Pi*(k + 1\/2))\/(n))",
"translationInformation": {
"subEquations": [
"x[k] = cos((Pi*(k + 1\/2))\/(n))"
],
"freeVariables": [
"k",
"n",
"x[k]"
],
"constraints": [
"k = 0 , .. , n - 1"
],
"tokenTranslations": {
"\\cos": "Cosine; Example: \\cos@@{z}\nWill be translated to: cos($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E2\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=cos",
"\\cpi": "Pi was translated to: Pi"
}
}
}
},
"positions": [
{
"section": 8,
"sentence": 2,
"word": 18
}
],
"includes": [
"n",
"x",
"n x",
"-1",
"k = 0",
"x_{k}"
],
"isPartOf": [],
"definiens": [
{
"definition": "root",
"score": 2
},
{
"definition": "one",
"score": 0
},
{
"definition": "trigonometric definition",
"score": 0
},
{
"definition": "fact",
"score": 0
},
{
"definition": "different simple root",
"score": 1
},
{
"definition": "Chebyshev polynomial of the first kind",
"score": 1
},
{
"definition": "Chebyshev polynomial",
"score": 1
}
]
}
]
}
|
| 13 | E(x, y; u) := \sum_{n=0}^\infty u^n \, \psi_n (x) \, \psi_n (y) = \frac{1}{\sqrt{\pi (1 - u^2)}} \, \exp\left(-\frac{1 - u}{1 + u} \, \frac{(x + y)^2}{4} - \frac{1 + u}{1 - u} \, \frac{(x - y)^2}{4}\right)
|
Hermite polynomials | Full data:
{
"id": 13,
"pid": 63,
"eid": "math.63.109",
"title": "Hermite polynomials",
"formulae": [
{
"id": "FORMULA_249043719eb4dd70350b460363255e11",
"formula": "E(x, y; u) := \\sum_{n=0}^\\infty u^n \\, \\psi_n (x) \\, \\psi_n (y) = \\frac{1}{\\sqrt{\\pi (1 - u^2)}} \\, \\exp\\left(-\\frac{1 - u}{1 + u} \\, \\frac{(x + y)^2}{4} - \\frac{1 + u}{1 - u} \\, \\frac{(x - y)^2}{4}\\right)",
"semanticFormula": "E(x , y ; u) : = \\sum_{n=0}^\\infty u^n \\psi_n(x) \\psi_n(y) = \\frac{1}{\\sqrt{\\cpi(1 - u^2)}} \\exp(- \\frac{1 - u}{1 + u} \\frac{(x + y)^2}{4} - \\frac{1 + u}{1 - u} \\frac{(x - y)^2}{4})",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "\\[CapitalEpsilon][x_, y_, u_] := Sum[(u)^(n)* Subscript[\\[Psi], n][x]* Subscript[\\[Psi], n][y], {n, 0, Infinity}, GenerateConditions->None] == Divide[1,Sqrt[Pi*(1 - (u)^(2))]]*Exp[-Divide[1 - u,1 + u]*Divide[(x + y)^(2),4]-Divide[1 + u,1 - u]*Divide[(x - y)^(2),4]]"
},
"Maple": {
"translation": "Epsilon := (x, y, u) -> sum((u)^(n)* psi[n](x)* psi[n](y), n = 0..infinity) = (1)\/(sqrt(Pi*(1 - (u)^(2))))*exp(-(1 - u)\/(1 + u)*((x + y)^(2))\/(4)-(1 + u)\/(1 - u)*((x - y)^(2))\/(4))"
}
},
"positions": [
{
"section": 25,
"sentence": 2,
"word": 16
}
],
"includes": [
"u",
"\\psi_{n}",
"H_{n}(x)",
"\\psi_{n}(x)",
"x^{n}",
"n",
"x",
"H_{n}",
"He_{n}(x)",
"He_{n}",
"D_{n}(z)",
"E(x,y;u)",
"H_{n}(y)"
],
"isPartOf": [],
"definiens": [
{
"definition": "distributional identity",
"score": 1
},
{
"definition": "separable kernel",
"score": 1
},
{
"definition": "Mehler 's formula",
"score": 2
},
{
"definition": "Hermite polynomial",
"score": 1
},
{
"definition": "Hermite function",
"score": 2
},
{
"definition": "Hermite",
"score": 1
},
{
"definition": "bivariate Gaussian probability density",
"score": 1
},
{
"definition": "Gaussian probability density",
"score": 1
},
{
"definition": "Gaussian probability",
"score": 1
}
]
}
]
}
|
| 14 | x = \pm 1
|
Legendre function | Full data:
{
"id": 14,
"pid": 64,
"eid": "math.64.8",
"title": "Legendre function",
"formulae": [
{
"id": "FORMULA_06f9b7b1d3f141742ad1c582b55056ba",
"formula": "x = \\pm 1",
"semanticFormula": "x = \\pm 1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "x == \\[PlusMinus]1",
"translationInformation": {
"subEquations": [
"x = + 1",
"x = - 1"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"\\pm": "was translated to: \\[PlusMinus]"
}
}
},
"Maple": {
"translation": "x = &+- 1",
"translationInformation": {
"subEquations": [
"x = + 1",
"x = - 1"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"\\pm": "was translated to: &+-"
}
}
}
},
"positions": [
{
"section": 3,
"sentence": 1,
"word": 11
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| 15 | E_n=2^nE_n(\tfrac{1}{2})
|
Bernoulli polynomials | Full data:
{
"id": 15,
"pid": 65,
"eid": "math.65.27",
"title": "Bernoulli polynomials",
"formulae": [
{
"id": "FORMULA_a7fcf738c676932d58f39ff9f7df22ae",
"formula": "E_n=2^nE_n(\\tfrac{1}{2})",
"semanticFormula": "\\EulernumberE{n} = 2^n\\EulerpolyE{n}@{\\tfrac{1}{2}}",
"confidence": 0.8953028732079359,
"translations": {
"Mathematica": {
"translation": "EulerE[n] == (2)^(n)* EulerE[n, Divide[1,2]]"
},
"Maple": {
"translation": "euler(n) = (2)^(n)* euler(n, (1)\/(2))"
}
},
"positions": [
{
"section": 8,
"sentence": 4,
"word": 6
}
],
"includes": [
"B_{n}",
"n",
"E_{k}"
],
"isPartOf": [],
"definiens": [
{
"definition": "Euler number",
"score": 2
}
]
}
]
}
|
| 16 | \operatorname{Si}(ix) = i\operatorname{Shi}(x)
|
Trigonometric integral | Full data:
{
"id": 16,
"pid": 66,
"eid": "math.66.8",
"title": "Trigonometric integral",
"formulae": [
{
"id": "FORMULA_0feb8031b89a9707b164163ec50265f0",
"formula": "\\operatorname{Si}(ix) = i\\operatorname{Shi}(x)",
"semanticFormula": "\\sinint@{\\iunit x} = \\iunit \\sinhint@{x}",
"confidence": 0.8811682126384021,
"translations": {
"Mathematica": {
"translation": "SinIntegral[I*x] == I*SinhIntegral[x]",
"translationInformation": {
"subEquations": [
"SinIntegral[I*x] == I*SinhIntegral[x]"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"Shi": "Was interpreted as a function call because of a leading \\operatorname.",
"\\iunit": "Imaginary unit was translated to: I",
"\\sinint": "Sine integral; Example: \\sinint@{z}\nWill be translated to: SinIntegral[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/6.2#E9\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/SinIntegral.html"
}
}
},
"Maple": {
"translation": "Si(I*x) = I*Shi(x)",
"translationInformation": {
"subEquations": [
"Si(I*x) = I*Shi(x)"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"Shi": "Was interpreted as a function call because of a leading \\operatorname.",
"\\iunit": "Imaginary unit was translated to: I",
"\\sinint": "Sine integral; Example: \\sinint@{z}\nWill be translated to: Si($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/6.2#E9\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=Si"
}
}
}
},
"positions": [
{
"section": 3,
"sentence": 1,
"word": 9
}
],
"includes": [
"Si",
"Si(x)",
"x"
],
"isPartOf": [],
"definiens": [
{
"definition": "ordinary sine",
"score": 1
},
{
"definition": "Trigonometric integral",
"score": 2
},
{
"definition": "hyperbolic sine integral",
"score": 2
}
]
}
]
}
|
| 17 | f(z)=\frac{1}{\Beta(x,y)}
|
Beta function | Full data:
{
"id": 17,
"pid": 67,
"eid": "math.67.29",
"title": "Beta function",
"formulae": [
{
"id": "FORMULA_5f59825d73d63a9990498edca7222261",
"formula": "f(z)=\\frac{1}{\\Beta(x,y)}",
"semanticFormula": "f(x, y) = \\frac{1}{\\EulerBeta@{x}{y}}",
"confidence": 0.8953028732079359,
"translations": {
"Mathematica": {
"translation": "f[x_, y_] := Divide[1,Beta[x, y]]"
},
"Maple": {
"translation": "f := (x,y) -> (1)\/(Beta(x, y))"
}
},
"positions": [
{
"section": 6,
"sentence": 0,
"word": 12
}
],
"includes": [
"x, y",
"\\Beta",
"y",
"x"
],
"isPartOf": [],
"definiens": [
{
"definition": "function about the form",
"score": 0
},
{
"definition": "reciprocal beta function",
"score": 2
},
{
"definition": "definite integral of trigonometric function",
"score": 1
},
{
"definition": "integral representation",
"score": 0
},
{
"definition": "product",
"score": 0
},
{
"definition": "power",
"score": 0
},
{
"definition": "multiple-angle",
"score": 0
},
{
"definition": "beta function",
"score": 2
}
]
}
]
}
|
| 18 | \begin{align}\int x^m e^{ix^n}\,dx & =\frac{x^{m+1}}{m+1}\,_1F_1\left(\begin{array}{c} \frac{m+1}{n}\\1+\frac{m+1}{n}\end{array}\mid ix^n\right) \\[6px]& =\frac{1}{n} i^\frac{m+1}{n}\gamma\left(\frac{m+1}{n},-ix^n\right),\end{align}
|
Fresnel integral | Full data:
{
"id": 18,
"pid": 68,
"eid": "math.68.51",
"title": "Fresnel integral",
"formulae": [
{
"id": "FORMULA_b7dae135f3b04317078f86b595fe7dae",
"formula": "\\begin{align}\\int x^m e^{ix^n}\\,dx & =\\frac{x^{m+1}}{m+1}\\,_1F_1\\left(\\begin{array}{c} \\frac{m+1}{n}\\\\1+\\frac{m+1}{n}\\end{array}\\mid ix^n\\right) \\\\[6px]& =\\frac{1}{n} i^\\frac{m+1}{n}\\gamma\\left(\\frac{m+1}{n},-ix^n\\right),\\end{align}",
"semanticFormula": "\\begin{align}\\int x^m \\exp(\\iunit x^n) \\diff{x} &= \\frac{x^{m+1}}{m+1}\\genhyperF{1}{1}@{\\frac{m+1}{n}}{1+\\frac{m+1}{n}}{\\iunit x^n}\\\\ &=\\frac{1}{n} \\iunit^{(m+1)\/n} \\incgamma@{\\frac{m+1}{n}}{-\\iunit x^n}\\end{align}",
"confidence": 0.869061849326977,
"translations": {
"Mathematica": {
"translation": "Integrate[(x)^(m)* Exp[I*(x)^(n)], x, GenerateConditions->None] == Divide[(x)^(m + 1),m + 1]*HypergeometricPFQ[{Divide[m + 1,n]}, {1 +Divide[m + 1,n]}, I*(x)^(n)] == Divide[1,n]*(I)^((m + 1)\/n)* Gamma[Divide[m + 1,n], 0, - I*(x)^(n)]"
},
"Maple": {
"translation": "int((x)^(m)* exp(I*(x)^(n)), x) = ((x)^(m + 1))\/(m + 1)*hypergeom([(m + 1)\/(n)], [1 +(m + 1)\/(n)], I*(x)^(n)) = (1)\/(n)*(I)^((m + 1)\/n)* GAMMA((m + 1)\/(n))-GAMMA((m + 1)\/(n), - I*(x)^(n))"
}
},
"positions": [
{
"section": 5,
"sentence": 0,
"word": 14
}
],
"includes": [
"dx",
"x"
],
"isPartOf": [],
"definiens": [
{
"definition": "incomplete gamma function",
"score": 2
},
{
"definition": "confluent hypergeometric function",
"score": 2
},
{
"definition": "Fresnel integral",
"score": 1
},
{
"definition": "imaginary part",
"score": 1
}
]
}
]
}
|
| 19 | T_n(x) = \frac{\Gamma(1/2)\sqrt{1-x^2}}{(-2)^n\,\Gamma(n+1/2)} \ \frac{d^n}{dx^n}\left([1-x^2]^{n-1/2}\right)
|
Classical orthogonal polynomials | Full data:
{
"id": 19,
"pid": 69,
"eid": "math.69.117",
"title": "Classical orthogonal polynomials",
"formulae": [
{
"id": "FORMULA_725c6b6b645d425d3b385ac2c002da77",
"formula": "T_n(x) = \\frac{\\Gamma(1\/2)\\sqrt{1-x^2}}{(-2)^n\\,\\Gamma(n+1\/2)} \\ \\frac{d^n}{dx^n}\\left([1-x^2]^{n-1\/2}\\right)",
"semanticFormula": "\\ChebyshevpolyT{n}@{x} = \\frac{\\EulerGamma{1\/2}\\sqrt{1-x^2}}{(-2)^n\\EulerGamma{n+1\/2}} \\deriv [n]{ }{x}([1 - x^2]^{n-1\/2})",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "ChebyshevT[n, x] == Divide[Gamma[1\/2]*Sqrt[1 - (x)^(2)],(- 2)^(n)* Gamma[n + 1\/2]]*D[(1 - (x)^(2))^(n - 1\/2), {x, n}]"
},
"Maple": {
"translation": "ChebyshevT(n, x) = (GAMMA(1\/2)*sqrt(1 - (x)^(2)))\/((- 2)^(n)* GAMMA(n + 1\/2))*diff((1 - (x)^(2))^(n - 1\/2), [x$(n)])"
}
},
"positions": [
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"sentence": 3,
"word": 4
}
],
"includes": [
"\\ L_n",
"H_n",
"P_{n}",
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"n",
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"P_n",
"\\lambda_{n}",
"-1",
"+1",
"U_n"
],
"isPartOf": [],
"definiens": [
{
"definition": "Rodrigues ' formula",
"score": 2
},
{
"definition": "orthogonal polynomial",
"score": 1
},
{
"definition": "Chebyshev polynomials of the second kind",
"score": 1
},
{
"definition": "classical orthogonal polynomial",
"score": 1
},
{
"definition": "Chebyshev polynomial",
"score": 2
},
{
"definition": "Gamma function",
"score": 2
}
]
}
]
}
|
| 20 | {}_1F_0(1;;z) = \sum_{n \geqslant 0} z^n = (1-z)^{-1}
|
Generalized hypergeometric function | Full data:
{
"id": 20,
"pid": 70,
"eid": "math.70.58",
"title": "Generalized hypergeometric function",
"formulae": [
{
"id": "FORMULA_699b5f465d21dd6af7212cd8414f60c6",
"formula": "{}_1F_0(1;;z) = \\sum_{n \\geqslant 0} z^n = (1-z)^{-1}",
"semanticFormula": "\\genhyperF{1}{0}@{1}{}{z} = \\sum_{n \\geqslant 0} z^n = (1-z)^{-1}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "HypergeometricPFQ[{1}, {}, z] == Sum[(z)^(n), {n, 0, Infinity}, GenerateConditions->None] == (1 - z)^(- 1)"
},
"Maple": {
"translation": "hypergeom([1], [], z) = sum((z)^(n), n = 0..infinity) = (1 - z)^(- 1)"
}
},
"positions": [
{
"section": 17,
"sentence": 2,
"word": 0
}
],
"includes": [
"z",
"n",
"z)",
"_{p}F_{q}",
"^{n}"
],
"isPartOf": [],
"definiens": [
{
"definition": "geometric series with ratio",
"score": 2
},
{
"definition": "coefficient",
"score": 0
},
{
"definition": "hypergeometric function",
"score": 2
}
]
}
]
}
|
| 21 | \chi(-1) = 1
|
Dirichlet L-function | Full data:
{
"id": 21,
"pid": 71,
"eid": "math.71.1-1",
"title": "Dirichlet L-function",
"formulae": [
{
"id": "FORMULA_dcb9beab8f504cfc907c3165d24e5ad3",
"formula": "\\chi(-1) = 1",
"semanticFormula": "\\Dirichletchar@@{- 1}{k} = 1",
"confidence": 0.746792096089683,
"translations": {
"Mathematica": {
"translation": "DirichletCharacter[1, 1, -1] == 1"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 7
}
],
"includes": [
"\\chi"
],
"isPartOf": [
"a=\\begin{cases}0;&\\mbox{if }\\chi(-1)=1, \\\\ 1;&\\mbox{if }\\chi(-1)=-1,\\end{cases}"
],
"definiens": [
{
"definition": "primitive character",
"score": 2
},
{
"definition": "integer",
"score": 1
},
{
"definition": "only zero",
"score": 0
},
{
"definition": "Gamma function",
"score": 0
},
{
"definition": "symbol",
"score": 0
},
{
"definition": "functional equation",
"score": 0
},
{
"definition": "Gauss sum",
"score": 0
},
{
"definition": "Dirichlet character",
"score": 2
}
]
}
]
}
|
| 22 | \operatorname{Bi}'(z)\sim \frac{z^{\frac{1}{4}}e^{\frac{2}{3}z^{\frac{3}{2}}}}{\sqrt\pi\,}\left[ \sum_{n=0}^{\infty}\frac{1+6n}{1-6n} \dfrac{ \Gamma(n+\frac{5}{6})\Gamma(n+\frac{1}{6})\left(\frac{3}{4}\right)^n}{2\pi n! z^{3n/2}} \right]
|
Airy function | Full data:
{
"id": 22,
"pid": 72,
"eid": "math.72.15",
"title": "Airy function",
"formulae": [
{
"id": "FORMULA_3b2520d05d324290456841271e8d565b",
"formula": "\\operatorname{Bi}'(z)\\sim \\frac{z^{\\frac{1}{4}}e^{\\frac{2}{3}z^{\\frac{3}{2}}}}{\\sqrt\\pi\\,}\\left[ \\sum_{n=0}^{\\infty}\\frac{1+6n}{1-6n} \\dfrac{ \\Gamma(n+\\frac{5}{6})\\Gamma(n+\\frac{1}{6})\\left(\\frac{3}{4}\\right)^n}{2\\pi n! z^{3n\/2}} \\right]",
"semanticFormula": "\\AiryBi'@{z} \\sim \\frac{z^{\\frac{1}{4}} \\expe^{\\frac{2}{3}z^{\\frac{3}{2}}}}{\\sqrt{\\cpi}} [\\sum_{n=0}^{\\infty} \\frac{1+6n}{1-6n} \\frac{\\EulerGamma@{n + \\frac{5}{6}} \\EulerGamma@{n + \\frac{1}{6}}(\\frac{3}{4})^n{2 \\cpi n! z^{3n\/2}}}]",
"confidence": 0.6525418663370697,
"translations": {},
"positions": [
{
"section": 3,
"sentence": 9,
"word": 9
}
],
"includes": [
"z",
"z)",
"= 0"
],
"isPartOf": [],
"definiens": [
{
"definition": "z",
"score": 0
},
{
"definition": "asymptotic formula for Ai",
"score": 1
},
{
"definition": "Bi",
"score": 1
},
{
"definition": "asymptotic behaviour of the Airy function",
"score": 1
},
{
"definition": "Ai",
"score": 1
},
{
"definition": "cosine",
"score": 2
},
{
"definition": "definition of the Airy function",
"score": 1
},
{
"definition": "Airy function",
"score": 2
},
{
"definition": "Gamma function",
"score": 2
}
]
}
]
}
|
| 23 | F'(y)=1-2yF(y)
|
Dawson function | Full data:
{
"id": 23,
"pid": 73,
"eid": "math.73.41",
"title": "Dawson function",
"formulae": [
{
"id": "FORMULA_f6b555bd8ce626d90119ab5eafdaeff2",
"formula": "F'(y)=1-2yF(y)",
"semanticFormula": "\\DawsonsintF'@{y}=1-2y\\DawsonsintF@{y}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "D[DawsonF[y], {y, 1}] == 1 - 2*y*DawsonF[y]"
},
"Maple": {
"translation": "diff( dawson(y), y$(1) ) = 1 - 2*y*dawson(y)"
}
},
"positions": [
{
"section": 2,
"sentence": 9,
"word": 1
}
],
"includes": [
"y"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 0
},
{
"definition": "Dawson function",
"score": 2
}
]
}
]
}
|
| 24 | s\not =1
|
Hurwitz zeta function | Full data:
{
"id": 24,
"pid": 74,
"eid": "math.74.0-1",
"title": "Hurwitz zeta function",
"formulae": [
{
"id": "FORMULA_80a3608d4c2aae63f082861007c16c38",
"formula": "s\\not =1",
"semanticFormula": "s \\neq 1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "s \\[NotEqual] 1"
}
},
"positions": [
{
"section": 0,
"sentence": 2,
"word": 24
}
],
"includes": [
"s",
"1",
"\\not = 1"
],
"isPartOf": [],
"definiens": [
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| 25 | q = e^{i\pi\tau}
|
Theta function | Full data:
{
"id": 25,
"pid": 75,
"eid": "math.75.6-1",
"title": "Theta function",
"formulae": [
{
"id": "FORMULA_bfba6c35dbbcd8b89c6a29b1ffd6f517",
"formula": "q = e^{i\\pi\\tau}",
"semanticFormula": "q = \\expe^{\\iunit \\cpi \\tau}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "q == Exp[I*Pi*\\[Tau]]"
},
"Maple": {
"translation": "q = exp(I*Pi*tau)"
}
},
"positions": [
{
"section": 2,
"sentence": 0,
"word": 57
}
],
"includes": [
"\\tau",
"q",
"w = e^{\\pi iz}",
"q = e^{\\pi i\\tau}"
],
"isPartOf": [
"q = e^{\\pi i\\tau}",
"q = e^{2\\pi i\\tau}",
"\\theta_F (z)= \\sum_{m\\in \\Z^n} e^{2\\pi izF(m)}",
"\\hat{\\theta}_F (z) = \\sum_{k=0}^\\infty R_F(k) e^{2\\pi ikz}"
],
"definiens": [
{
"definition": "term of the nome",
"score": 2
},
{
"definition": "nome",
"score": 2
}
]
}
]
}
|
| 26 | \frac{\mathrm{d}}{\mathrm{d}z} \operatorname{dn}(z) = - k^2 \operatorname{sn}(z) \operatorname{cn}(z)
|
Jacobi elliptic functions | Full data:
{
"id": 26,
"pid": 76,
"eid": "math.76.155",
"title": "Jacobi elliptic functions",
"formulae": [
{
"id": "FORMULA_b54c03865b3efa9ea9112567cd66f59d",
"formula": "\\frac{\\mathrm{d}}{\\mathrm{d}z} \\operatorname{dn}(z) = - k^2 \\operatorname{sn}(z) \\operatorname{cn}(z)",
"semanticFormula": "\\deriv [1]{ }{z} \\Jacobielldnk@@{(z)}{k} = - k^2 \\Jacobiellsnk@@{(z)}{k} \\Jacobiellcnk@@{(z)}{k}",
"confidence": 0.6954186066124032,
"translations": {
"Mathematica": {
"translation": "D[JacobiDN[z, (k)^2], {z, 1}] == - (k)^(2)* JacobiSN[z, (k)^2]*JacobiCN[z, (k)^2]"
},
"Maple": {
"translation": "diff(JacobiDN(z, k), [z$(1)]) = - (k)^(2)* JacobiSN(z, k)*JacobiCN(z, k)"
}
},
"positions": [
{
"section": 16,
"sentence": 0,
"word": 15
}
],
"includes": [
"k",
"^{2}"
],
"isPartOf": [],
"definiens": [
{
"definition": "derivative",
"score": 2
},
{
"definition": "elliptic function",
"score": 2
},
{
"definition": "basic Jacobi",
"score": 0
},
{
"definition": "sn",
"score": 2
},
{
"definition": "dn",
"score": 2
},
{
"definition": "cn",
"score": 2
}
]
}
]
}
|
| 27 | \int_{-\infty}^\infty \frac {\gamma\left(\frac s 2, z^2 \pi \right)} {(z^2 \pi)^\frac s 2} e^{-2 \pi i k z} \mathrm d z = \frac {\Gamma\left(\frac {1-s} 2, k^2 \pi \right)} {(k^2 \pi)^\frac {1-s} 2}
|
Incomplete gamma function | Full data:
{
"id": 27,
"pid": 77,
"eid": "math.77.118",
"title": "Incomplete gamma function",
"formulae": [
{
"id": "FORMULA_c82b4ceebacd2b4a03b2eff406834e61",
"formula": "\\int_{-\\infty}^\\infty \\frac {\\gamma\\left(\\frac s 2, z^2 \\pi \\right)} {(z^2 \\pi)^\\frac s 2} e^{-2 \\pi i k z} \\mathrm d z = \\frac {\\Gamma\\left(\\frac {1-s} 2, k^2 \\pi \\right)} {(k^2 \\pi)^\\frac {1-s} 2}",
"semanticFormula": "\\int_{-\\infty}^\\infty \\frac{\\incgamma@{\\frac s 2}{z^2 \\cpi}}{(z^2 \\cpi)^\\frac s 2} \\expe^{- 2 \\cpi \\iunit k z} \\diff{z} = \\frac{\\incGamma@{\\frac {1-s} 2}{k^2 \\cpi}}{(k^2 \\cpi)^\\frac {1-s} 2}}",
"confidence": 0.8121295595054496,
"translations": {
"Mathematica": {
"translation": "Integrate[Divide[Gamma[Divide[s,2], 0, (z)^(2)* Pi],((z)^(2)* Pi)^(Divide[s,2])]*Exp[- 2*Pi*I*k*z], {z, - Infinity, Infinity}, GenerateConditions->None] == Divide[Gamma[Divide[1 - s,2], (k)^(2)* Pi],((k)^(2)* Pi)^(Divide[1 - s,2])]"
},
"Maple": {
"translation": "int((GAMMA((s)\/(2))-GAMMA((s)\/(2), (z)^(2)* Pi))\/(((z)^(2)* Pi)^((s)\/(2)))*exp(- 2*Pi*I*k*z), z = - infinity..infinity) = (GAMMA((1 - s)\/(2), (k)^(2)* Pi))\/(((k)^(2)* Pi)^((1 - s)\/(2)))"
}
},
"positions": [
{
"section": 25,
"sentence": 1,
"word": 15
}
],
"includes": [
"\\gamma(s, z)",
"\\gamma",
"z^s",
"\\Gamma",
"\\gamma(s,z)",
"k",
"z",
"z=",
"2\\pi",
"\\gamma(u,v)",
"\\gamma(s,x)",
"s",
"z^{s}",
"e^{-x}",
"\\gamma(a,x)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Fourier",
"score": 1
},
{
"definition": "upper incomplete Gamma function",
"score": 2
},
{
"definition": "lower incomplete Gamma function",
"score": 2
}
]
}
]
}
|
| 28 | _{1}(z) =
|
Polylogarithm | Full data:
{
"id": 28,
"pid": 78,
"eid": "math.78.0-1",
"title": "Polylogarithm",
"formulae": [
{
"id": "FORMULA_e939f30d07578c2fb0d8cb5201db3c79",
"formula": "_{1}(z) =",
"semanticFormula": "\\polylog{1}@{z} = -\\ln@{1-z}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "PolyLog[1, z] = -Log[1 - z]"
},
"Maple": {
"translation": "polylog(1, z) = -ln(1 - z)"
}
},
"positions": [
{
"section": 0,
"sentence": 7,
"word": 11
}
],
"includes": [
"_{1}",
"z",
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"z)",
"1"
],
"isPartOf": [
"\\operatorname{Li}_{1}(z) = -\\ln(1-z)",
"\\operatorname{Ti}_0(z) = {z \\over 1+z^2}, \\quad \\operatorname{Ti}_1(z) = \\arctan z, \\quad \\operatorname{Ti}_2(z) = \\int_0^z {\\arctan t \\over t} dt, \\quad \\ldots\\quad \\operatorname{Ti}_{n+1}(z) = \\int_0^z \\frac{\\operatorname{Ti}_n(t)}{t} dt"
],
"definiens": [
{
"definition": "natural logarithm",
"score": 2
},
{
"definition": "logarithm",
"score": 2
},
{
"definition": "polylogarithm function",
"score": 2
},
{
"definition": "dilogarithm",
"score": 1
},
{
"definition": "trilogarithm",
"score": 1
}
]
}
]
}
|
| 29 | \int_{-\infty}^\infty \operatorname{sinc}(t) \, e^{-i 2 \pi f t}\,dt = \operatorname{rect}(f)
|
Sinc function | Full data:
{
"id": 29,
"pid": 79,
"eid": "math.79.11",
"title": "Sinc function",
"formulae": [
{
"id": "FORMULA_6340f4a043f912a3557e084aaf03792a",
"formula": "\\int_{-\\infty}^\\infty \\operatorname{sinc}(t) \\, e^{-i 2 \\pi f t}\\,dt = \\operatorname{rect}(f)",
"semanticFormula": "\\int_{-\\infty}^\\infty \\operatorname{sinc}(t) \\expe^{- \\iunit 2 \\cpi f t} \\diff{t} = \\operatorname{rect}(f)",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Integrate[sinc[(t)]*Exp[- I*2*Pi*f*t], {t, - Infinity, Infinity}, GenerateConditions->None] == rect[f]"
},
"Maple": {
"translation": "int(sinc((t))*exp(- I*2*Pi*f*t), t = - infinity..infinity) = rect(f)"
}
},
"positions": [
{
"section": 1,
"sentence": 9,
"word": 16
}
],
"includes": [
"\\pi",
"\\infty",
"sinc"
],
"isPartOf": [],
"definiens": [
{
"definition": "argument",
"score": 0
},
{
"definition": "continuous Fourier",
"score": 2
},
{
"definition": "rectangular function",
"score": 2
},
{
"definition": "sinc",
"score": 2
},
{
"definition": "ordinary frequency",
"score": 1
}
]
}
]
}
|
| 30 | N=1
|
Exponential integral | Full data:
{
"id": 30,
"pid": 80,
"eid": "math.80.26",
"title": "Exponential integral",
"formulae": [
{
"id": "FORMULA_a9a738ef9d4e46360dd9b87b39c691bf",
"formula": "N=1",
"semanticFormula": "N=1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "N == 1"
},
"Maple": {
"translation": "N = 1"
}
},
"positions": [
{
"section": 4,
"sentence": 3,
"word": 30
}
],
"includes": [
"N"
],
"isPartOf": [],
"definiens": [
{
"definition": "large value",
"score": 2
},
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| 31 | \sum_{n=0}^\infty \frac{n!\,\Gamma\left(\alpha + 1\right)}{\Gamma\left(n+\alpha+1\right)}L_n^{(\alpha)}(x)L_n^{(\alpha)}(y)t^n=\frac{1}{(1-t)^{\alpha + 1}}e^{-(x+y)t/(1-t)}\,_0F_1\left(;\alpha + 1;\frac{xyt}{(1-t)^2}\right)
|
Laguerre polynomials | Full data:
{
"id": 31,
"pid": 81,
"eid": "math.81.84",
"title": "Laguerre polynomials",
"formulae": [
{
"id": "FORMULA_f179a85d8102cbedb67cf60b188a68b7",
"formula": "\\sum_{n=0}^\\infty \\frac{n!\\,\\Gamma\\left(\\alpha + 1\\right)}{\\Gamma\\left(n+\\alpha+1\\right)}L_n^{(\\alpha)}(x)L_n^{(\\alpha)}(y)t^n=\\frac{1}{(1-t)^{\\alpha + 1}}e^{-(x+y)t\/(1-t)}\\,_0F_1\\left(;\\alpha + 1;\\frac{xyt}{(1-t)^2}\\right)",
"semanticFormula": "\\sum_{n=0}^\\infty \\frac{n! \\EulerGamma@{\\alpha + 1}}{\\EulerGamma@{n + \\alpha + 1}} \\LaguerrepolyL[\\alpha]{n}@{x} \\LaguerrepolyL[\\alpha]{n}@{x} t^n = \\frac{1}{(1-t)^{\\alpha + 1}} \\expe^{-(x+y)t\/(1-t)} \\genhyperF{0}{1}@{}{\\alpha + 1}{\\frac{xyt}{(1-t)^2}}",
"confidence": 0.8953028732079359,
"translations": {
"Mathematica": {
"translation": "Sum[Divide[(n)!*Gamma[\\[Alpha]+ 1],Gamma[n + \\[Alpha]+ 1]]*LaguerreL[n, \\[Alpha], x]*LaguerreL[n, \\[Alpha], x]*(t)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,(1 - t)^(\\[Alpha]+ 1)]*Exp[-(x + y)*t\/(1 - t)]*HypergeometricPFQ[{}, {\\[Alpha]+ 1}, Divide[x*y*t,(1 - t)^(2)]]"
},
"Maple": {
"translation": "sum((factorial(n)*GAMMA(alpha + 1))\/(GAMMA(n + alpha + 1))*LaguerreL(n, alpha, x)*LaguerreL(n, alpha, x)*(t)^(n), n = 0..infinity) = (1)\/((1 - t)^(alpha + 1))*exp(-(x + y)*t\/(1 - t))*hypergeom([], [alpha + 1], (x*y*t)\/((1 - t)^(2)))"
}
},
"positions": [
{
"section": 15,
"sentence": 0,
"word": 10
}
],
"includes": [
"\\alpha",
"L_{n}^{(\\alpha)}",
"L_n^{(\\alpha)}(x)",
"n"
],
"isPartOf": [],
"definiens": [
{
"definition": "Hille formula",
"score": 2
},
{
"definition": "Laguerre polynomial",
"score": 2
},
{
"definition": "series on the left converge",
"score": 0
},
{
"definition": "generalized Laguerre polynomial",
"score": 2
},
{
"definition": "confluent hypergeometric function",
"score": 2
}
]
}
]
}
|
| 32 | c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!}
|
Associated Legendre polynomials | Full data:
{
"id": 32,
"pid": 82,
"eid": "math.82.8",
"title": "Associated Legendre polynomials",
"formulae": [
{
"id": "FORMULA_6f29e15c07089506a70db1b3f54b27a5",
"formula": "c_{lm} = (-1)^m \\frac{(\\ell-m)!}{(\\ell+m)!}",
"semanticFormula": "c_{lm} = (-1)^m \\frac{(\\ell-m)!}{(\\ell+m)!}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Subscript[c, l, m] == (- 1)^(m)*Divide[(\\[ScriptL]- m)!,(\\[ScriptL]+ m)!]"
},
"Maple": {
"translation": "c[l, m] = (- 1)^(m)*(factorial(ell - m))\/(factorial(ell + m))"
}
},
"positions": [
{
"section": 1,
"sentence": 7,
"word": 26
}
],
"includes": [
"m",
"(-1)^{m}",
"- 1"
],
"isPartOf": [],
"definiens": [
{
"definition": "proportionality constant",
"score": 2
}
]
}
]
}
|
| 33 | \mathrm{Gi}(x) = \frac{1}{\pi} \int_0^\infty \sin\left(\frac{t^3}{3} + xt\right)\, dt
|
Scorer's function | Full data:
{
"id": 33,
"pid": 83,
"eid": "math.83.3",
"title": "Scorer's function",
"formulae": [
{
"id": "FORMULA_c8116180276232704ca3e9f67f207565",
"formula": "\\mathrm{Gi}(x) = \\frac{1}{\\pi} \\int_0^\\infty \\sin\\left(\\frac{t^3}{3} + xt\\right)\\, dt",
"semanticFormula": "\\ScorerGi@{x} = \\frac{1}{\\cpi} \\int_0^\\infty \\sin(\\frac{t^3}{3} + xt) \\diff{t}",
"confidence": 0.7929614010341081,
"translations": {
"Mathematica": {
"translation": "ScorerGi[x] == Divide[1,Pi]*Integrate[Sin[Divide[(t)^(3),3]+ x*t], {t, 0, Infinity}, GenerateConditions->None]",
"translationInformation": {
"subEquations": [
"ScorerGi[x] = Divide[1,Pi]*Integrate[Sin[Divide[(t)^(3),3]+ x*t], {t, 0, Infinity}, GenerateConditions->None]"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"\\ScorerGi": "Scorer function Gi; Example: \\ScorerGi@{z}\nWill be translated to: ScorerGi[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/9.12#i\nMathematica: https:\/\/",
"\\cpi": "Pi was translated to: Pi",
"\\sin": "Sine; Example: \\sin@@{z}\nWill be translated to: Sin[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E1\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Sin.html"
}
}
},
"Maple": {
"translation": "AiryBi(x)*(int(AiryAi(t), t = (x) .. infinity))+AiryAi(x)*(int(AiryBi(t), t = 0 .. (x))) = (1)\/(Pi)*int(sin(((t)^(3))\/(3)+ x*t), t = 0..infinity)",
"translationInformation": {
"subEquations": [
"AiryBi(x)*(int(AiryAi(t), t = (x) .. infinity))+AiryAi(x)*(int(AiryBi(t), t = 0 .. (x))) = (1)\/(Pi)*int(sin(((t)^(3))\/(3)+ x*t), t = 0..infinity)"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"\\ScorerGi": "Scorer function Gi; Example: \\ScorerGi@{z}\nWill be translated to: \nAlternative translations: [AiryBi($0)*(int(AiryAi(t), t = ($0) .. infinity))+AiryAi($0)*(int(AiryBi(t), t = 0 .. ($0)))]Relevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/9.12#i\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=Airy",
"\\cpi": "Pi was translated to: Pi",
"\\sin": "Sine; Example: \\sin@@{z}\nWill be translated to: sin($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E1\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=sin"
}
}
}
},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 15
}
],
"includes": [
"x",
"x)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Scorer 's function",
"score": 2
},
{
"definition": "special function",
"score": 1
}
]
}
]
}
|
| 34 | \frac{\partial^2}{\partial x^2} V(x;\sigma,\gamma)= \frac{x^2-\gamma^2-\sigma^2}{\sigma^4} \frac{\operatorname{Re}[w(z)]}{\sigma\sqrt{2 \pi}}-\frac{2 x \gamma}{\sigma^4} \frac{\operatorname{Im}[w(z)]}{\sigma\sqrt{2 \pi}}+\frac{\gamma}{\sigma^4}\frac{1}{\pi}
|
Voigt profile | Full data:
{
"id": 34,
"pid": 84,
"eid": "math.84.31",
"title": "Voigt profile",
"formulae": [
{
"id": "FORMULA_e663d20df3cca1ae5dec645d320cd511",
"formula": "\\frac{\\partial^2}{\\partial x^2} V(x;\\sigma,\\gamma)= \\frac{x^2-\\gamma^2-\\sigma^2}{\\sigma^4} \\frac{\\operatorname{Re}[w(z)]}{\\sigma\\sqrt{2 \\pi}}-\\frac{2 x \\gamma}{\\sigma^4} \\frac{\\operatorname{Im}[w(z)]}{\\sigma\\sqrt{2 \\pi}}+\\frac{\\gamma}{\\sigma^4}\\frac{1}{\\pi}",
"semanticFormula": "\\deriv[2]{}{x} V(x ; \\sigma , \\gamma) = \\frac{x^2-\\gamma^2-\\sigma^2}{\\sigma^4} \\frac{\\realpart [\\Faddeevaw@{z}]}{\\sigma \\sqrt{2 \\cpi}} - \\frac{2 x \\gamma}{\\sigma^4} \\frac{\\imagpart [\\Faddeevaw@{z}]}{\\sigma \\sqrt{2 \\cpi}} + \\frac{\\gamma}{\\sigma^4} \\frac{1}{\\cpi}",
"confidence": 0.8620216359266987,
"translations": {
"Mathematica": {
"translation": "D[PDF[VoigtDistribution[\\[Gamma], \\[Sigma]], x], {x, 2}] == Divide[x^2 - \\[Gamma]^2 - \\[Sigma]^2, \\[Sigma]^4] * Divide[ Re[ Exp[-(Divide[x+I*y,\\[Sigma]*Sqrt[2]])^2]*Erfc[-I*(Divide[x+I*y,\\[Sigma]*Sqrt[2]])] ], \\[Sigma]*Sqrt[2*Pi]] - Divide[2*x*y, \\[Sigma]^4] * Divide[Im[Exp[-(Divide[x+I*y,\\[Sigma]*Sqrt[2]])^2]*Erfc[-I*(Divide[x+I*y,\\[Sigma]*Sqrt[2]])]], \\[Sigma]*Sqrt[2*Pi]] + Divide[\\[Gamma],\\[Sigma]^4]*Divide[1,Pi]"
}
},
"positions": [
{
"section": 6,
"sentence": 0,
"word": 20
}
],
"includes": [
"w(z)]",
"z",
"V(x;\\sigma,\\gamma)",
"x",
"w(z)"
],
"isPartOf": [],
"definiens": [
{
"definition": "term of the Faddeeva function",
"score": 2
},
{
"definition": "second derivative profile",
"score": 2
},
{
"definition": "real part of the Faddeeva function",
"score": 2
},
{
"definition": "Faddeeva function",
"score": 2
},
{
"definition": "Voigt function",
"score": 2
},
{
"definition": "Voigt profile",
"score": 2
}
]
}
]
}
|
| 35 | \Phi(z,s,a) = \frac{1}{1-z} \frac{1}{a^{s}} + \sum_{n=1}^{N-1} \frac{(-1)^{n} \mathrm{Li}_{-n}(z)}{n!} \frac{(s)_{n}}{a^{n+s}} +O(a^{-N-s})
|
Lerch zeta function | Full data:
{
"id": 35,
"pid": 85,
"eid": "math.85.57",
"title": "Lerch zeta function",
"formulae": [
{
"id": "FORMULA_a0cc62efe3cabac6d8bebe5b8b94b5fa",
"formula": "\\Phi(z,s,a) = \\frac{1}{1-z} \\frac{1}{a^{s}} + \\sum_{n=1}^{N-1} \\frac{(-1)^{n} \\mathrm{Li}_{-n}(z)}{n!} \\frac{(s)_{n}}{a^{n+s}} +O(a^{-N-s})",
"semanticFormula": "\\Phi(z , s , a) = \\frac{1}{1-z} \\frac{1}{a^{s}} + \\sum_{n=1}^{N-1} \\frac{(-1)^{n} \\polylog{-n}@{z}}{n!} \\frac{\\Pochhammersym{s}{n}}{a^{n+s}} + \\bigO{a^{-N-s}}",
"confidence": 0.8662724998444776,
"translations": {
"Mathematica": {
"translation": "\\[CapitalPhi][z, s, a] == Divide[1,1 - z]*Divide[1,(a)^(s)]+ Sum[Divide[(- 1)^(n)* PolyLog[-n, z],(n)!]*Divide[Pochhammer[s, n],(a)^(n + s)], {n, 1, N - 1}, GenerateConditions->None]+ O[a]^(- N - s)"
}
},
"positions": [
{
"section": 6,
"sentence": 1,
"word": 23
}
],
"includes": [
"a",
"\\Phi(z,s,a)",
"z",
"s"
],
"isPartOf": [],
"definiens": [
{
"definition": "asymptotic expansion",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "Lerch transcendent",
"score": 2
},
{
"definition": "polylogarithm",
"score": 2
},
{
"definition": "polylogarithm function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 2
}
]
}
]
}
|
| 36 | M(1,2,z)=(e^z-1)/z,\ \ M(1,3,z)=2!(e^z-1-z)/z^2
|
Confluent hypergeometric function | Full data:
{
"id": 36,
"pid": 86,
"eid": "math.86.44",
"title": "Confluent hypergeometric function",
"formulae": [
{
"id": "FORMULA_d83a3ce5244b566d8f71edb7f81afa43",
"formula": "M(1,2,z)=(e^z-1)\/z,\\ \\ M(1,3,z)=2!(e^z-1-z)\/z^2",
"semanticFormula": "\\KummerconfhyperM@{1}{2}{z} = (\\expe^z - 1) \/ z , \\KummerconfhyperM@{1}{3}{z} = 2! (\\expe^z - 1 - z) \/ z^2",
"confidence": 0.912945064646862,
"translations": {
"Mathematica": {
"translation": "Hypergeometric1F1[1, 2, z] == (Exp[z]- 1)\/z\n Hypergeometric1F1[1, 3, z] == (2)!*(Exp[z]- 1 - z)\/(z)^(2)"
},
"Maple": {
"translation": "KummerM(1, 2, z) = (exp(z)- 1)\/z; KummerM(1, 3, z) = factorial(2)*(exp(z)- 1 - z)\/(z)^(2)"
}
},
"positions": [
{
"section": 10,
"sentence": 4,
"word": 0
}
],
"includes": [
"M",
"U(a, b, z)",
"z",
"U(n,c,z)",
"\\Phi(a, b, z)",
"M(n,b,z)",
"M(a, b, z)"
],
"isPartOf": [],
"definiens": [
{
"definition": "etc",
"score": 0
},
{
"definition": "Kummer 's function of the first kind",
"score": 2
},
{
"definition": "confluent hypergeometric function",
"score": 1
},
{
"definition": "hypergeometric function",
"score": 1
}
]
}
]
}
|
| 37 | \sigma = \pm 1
|
Mathieu function | Full data:
{
"id": 37,
"pid": 87,
"eid": "math.87.54",
"title": "Mathieu function",
"formulae": [
{
"id": "FORMULA_f694135eafc20195a9d96ca3ce8af674",
"formula": "\\sigma = \\pm 1",
"semanticFormula": "\\sigma = \\pm 1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "\\[Sigma] == \\[PlusMinus]1",
"translationInformation": {
"subEquations": [
"\\[Sigma] = + 1",
"\\[Sigma] = - 1"
],
"freeVariables": [
"\\[Sigma]"
],
"constraints": [],
"tokenTranslations": {
"\\pm": "was translated to: \\[PlusMinus]"
}
}
},
"Maple": {
"translation": "sigma = &+- 1",
"translationInformation": {
"subEquations": [
"sigma = + 1",
"sigma = - 1"
],
"freeVariables": [
"sigma"
],
"constraints": [],
"tokenTranslations": {
"\\pm": "was translated to: &+-"
}
}
}
},
"positions": [
{
"section": 4,
"sentence": 1,
"word": 27
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| 38 | \frac{d^2f}{dz^2} + \left(\tilde{a}z^2+\tilde{b}z+\tilde{c}\right)f=0
|
Parabolic cylinder function | Full data:
{
"id": 38,
"pid": 88,
"eid": "math.88.0",
"title": "Parabolic cylinder function",
"formulae": [
{
"id": "FORMULA_bec6388631b20f2af14e375b13e1533f",
"formula": "\\frac{d^2f}{dz^2} + \\left(\\tilde{a}z^2+\\tilde{b}z+\\tilde{c}\\right)f=0",
"semanticFormula": "\\deriv [2]{f}{z} +(\\tilde{a} z^2 + \\tilde{b} z + \\tilde{c}) f = 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "D[f[z], {z, 2}] + (a*z^2 + b*z + c)*f[z] == 0"
}
},
"positions": [
{
"section": 0,
"sentence": 0,
"word": 19
}
],
"includes": [
"z"
],
"isPartOf": [],
"definiens": [
{
"definition": "solution to the differential equation",
"score": 2
},
{
"definition": "special function",
"score": 1
},
{
"definition": "mathematics",
"score": 0
},
{
"definition": "parabolic cylinder function",
"score": 1
}
]
}
]
}
|
| 39 | c=\infty
|
Painlevé transcendents | Full data:
{
"id": 39,
"pid": 89,
"eid": "math.89.23",
"title": "Painlev\u00e9 transcendents",
"formulae": [
{
"id": "FORMULA_0a306ab913684a1ba3935715d3dd8ad8",
"formula": "c=\\infty",
"semanticFormula": "c=\\infty",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "c == Infinity"
},
"Maple": {
"translation": "c = infinity"
}
},
"positions": [
{
"section": 9,
"sentence": 5,
"word": 23
}
],
"includes": [
"c"
],
"isPartOf": [],
"definiens": [
{
"definition": "central charge of the Virasoro algebra",
"score": 2
},
{
"definition": "combination of conformal block",
"score": 1
},
{
"definition": "Painlev\u00e9 VI equation",
"score": 1
},
{
"definition": "two-dimensional conformal field theory",
"score": 1
}
]
}
]
}
|
| 40 | c = a + 1
|
Hypergeometric function | Full data:
{
"id": 40,
"pid": 90,
"eid": "math.90.7",
"title": "Hypergeometric function",
"formulae": [
{
"id": "FORMULA_aaffb0ad8dea17d68491d9fb6ebcfbe3",
"formula": "c = a + 1",
"semanticFormula": "c = a + 1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "c == a + 1"
},
"Maple": {
"translation": "c := a + 1"
}
},
"positions": [
{
"section": 3,
"sentence": 0,
"word": 20
}
],
"includes": [
"a",
"c"
],
"isPartOf": [],
"definiens": [
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| 41 | \frac{1}{\Gamma(z)}= z e^{\gamma z} \prod_{k=1}^\infty \left\{ \left(1+\frac{z}{k}\right)e^{-z/k} \right\}
|
Barnes G-function | Full data:
{
"id": 41,
"pid": 91,
"eid": "math.91.47",
"title": "Barnes G-function",
"formulae": [
{
"id": "FORMULA_6bc0d742c4d25c1abb61158150489676",
"formula": "\\frac{1}{\\Gamma(z)}= z e^{\\gamma z} \\prod_{k=1}^\\infty \\left\\{ \\left(1+\\frac{z}{k}\\right)e^{-z\/k} \\right\\}",
"semanticFormula": "\\frac{1}{\\EulerGamma@{z}} = z \\expe^{\\EulerConstant z} \\prod_{k=1}^\\infty \\{(1 + \\frac{z}{k}) \\expe^{-z\/k} \\}",
"confidence": 0.8614665289982916,
"translations": {
"Mathematica": {
"translation": "Divide[1,Gamma[z]] == z*Exp[EulerGamma*z]*Product[(1 +Divide[z,k])*Exp[- z\/k], {k, 1, Infinity}, GenerateConditions->None]",
"translationInformation": {
"subEquations": [
"Divide[1,Gamma[z]] = z*Exp[EulerGamma*z]*Product[(1 +Divide[z,k])*Exp[- z\/k], {k, 1, Infinity}, GenerateConditions->None]"
],
"freeVariables": [
"z"
],
"constraints": [],
"tokenTranslations": {
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\EulerConstant": "Euler-Mascheroni constant was translated to: EulerGamma",
"\\EulerGamma": "Euler Gamma function; Example: \\EulerGamma@{z}\nWill be translated to: Gamma[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/5.2#E1\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Gamma.html"
}
}
},
"Maple": {
"translation": "(1)\/(GAMMA(z)) = z*exp(gamma*z)*product((1 +(z)\/(k))*exp(- z\/k), k = 1..infinity)",
"translationInformation": {
"subEquations": [
"(1)\/(GAMMA(z)) = z*exp(gamma*z)*product((1 +(z)\/(k))*exp(- z\/k), k = 1..infinity)"
],
"freeVariables": [
"z"
],
"constraints": [],
"tokenTranslations": {
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\EulerConstant": "Euler-Mascheroni constant was translated to: gamma",
"\\EulerGamma": "Euler Gamma function; Example: \\EulerGamma@{z}\nWill be translated to: GAMMA($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/5.2#E1\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=GAMMA"
}
}
}
},
"positions": [
{
"section": 8,
"sentence": 0,
"word": 55
}
],
"includes": [
"\\,\\Gamma(x)",
"\\, \\gamma",
"z",
"\\,\\gamma"
],
"isPartOf": [],
"definiens": [
{
"definition": "Euler",
"score": 1
},
{
"definition": "Mascheroni",
"score": 1
},
{
"definition": "gamma function",
"score": 2
}
]
}
]
}
|
| 42 | 192/24 = 8 = 2 \times 4
|
Heun function | Full data:
{
"id": 42,
"pid": 92,
"eid": "math.92.1-1",
"title": "Heun function",
"formulae": [
{
"id": "FORMULA_8c78ef87048e61947a6d7d4b5e06aa63",
"formula": "192\/24 = 8 = 2 \\times 4",
"semanticFormula": "192\/24 = 8 = 2 \\times 4",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "192\/24 == 8 == 2 * 4",
"translationInformation": {
"subEquations": [
"192\/24 = 8",
"8 = 2 * 4"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\times": "was translated to: *"
}
},
"numericResults": {
"overallResult": "SUCCESS",
"numberOfTests": 2,
"numberOfFailedTests": 0,
"numberOfSuccessfulTests": 2,
"numberOfSkippedTests": 0,
"numberOfErrorTests": 0,
"wasAborted": false,
"crashed": false,
"testCalculationsGroups": [
{
"lhs": "192\/24",
"rhs": "8",
"testExpression": "(192\/24)-(8)",
"activeConstraints": [],
"testCalculations": [
{
"result": "SUCCESS",
"resultExpression": "0.",
"testValues": {}
}
]
},
{
"lhs": "8",
"rhs": "2 * 4",
"testExpression": "(8)-(2 * 4)",
"activeConstraints": [],
"testCalculations": [
{
"result": "SUCCESS",
"resultExpression": "0.",
"testValues": {}
}
]
}
]
},
"symbolicResults": {
"overallResult": "SUCCESS",
"numberOfTests": 2,
"numberOfFailedTests": 0,
"numberOfSuccessfulTests": 2,
"numberOfSkippedTests": 0,
"numberOfErrorTests": 0,
"crashed": false,
"testCalculationsGroup": [
{
"lhs": "192\/24",
"rhs": "8",
"testExpression": "(192\/24)-(8)",
"testCalculations": [
{
"result": "SUCCESS",
"testTitle": "Simple",
"testExpression": "FullSimplify[(192\/24)-(8)]",
"resultExpression": "0",
"wasAborted": false,
"conditionallySuccessful": false
}
]
},
{
"lhs": "8",
"rhs": "2 * 4",
"testExpression": "(8)-(2 * 4)",
"testCalculations": [
{
"result": "SUCCESS",
"testTitle": "Simple",
"testExpression": "FullSimplify[(8)-(2 * 4)]",
"resultExpression": "0",
"wasAborted": false,
"conditionallySuccessful": false
}
]
}
]
}
},
"SymPy": {
"translation": "192\/24 == 8 == 2 * 4",
"translationInformation": {
"subEquations": [
"192\/24 = 8",
"8 = 2 * 4"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\times": "was translated to: *"
}
}
},
"Maple": {
"translation": "192\/24 = 8 = 2 * 4",
"translationInformation": {
"subEquations": [
"192\/24 = 8",
"8 = 2 * 4"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\times": "was translated to: *"
}
},
"numericResults": {
"overallResult": "SUCCESS",
"numberOfTests": 2,
"numberOfFailedTests": 0,
"numberOfSuccessfulTests": 2,
"numberOfSkippedTests": 0,
"numberOfErrorTests": 0,
"wasAborted": false,
"crashed": false,
"testCalculationsGroups": [
{
"lhs": "192\/24",
"rhs": "8",
"testExpression": "evalf((192\/24)-(8))",
"activeConstraints": [],
"testCalculations": [
{
"result": "SUCCESS",
"resultExpression": "0.",
"testValues": {}
}
]
},
{
"lhs": "8",
"rhs": "2 * 4",
"testExpression": "evalf((8)-(2 * 4))",
"activeConstraints": [],
"testCalculations": [
{
"result": "SUCCESS",
"resultExpression": "0.",
"testValues": {}
}
]
}
]
},
"symbolicResults": {
"overallResult": "SUCCESS",
"numberOfTests": 2,
"numberOfFailedTests": 0,
"numberOfSuccessfulTests": 2,
"numberOfSkippedTests": 0,
"numberOfErrorTests": 0,
"crashed": false,
"testCalculationsGroup": [
{
"lhs": "192\/24",
"rhs": "8",
"testExpression": "(192\/24)-(8)",
"testCalculations": [
{
"result": "SUCCESS",
"testTitle": "Simple",
"testExpression": "simplify((192\/24)-(8))",
"resultExpression": "0",
"wasAborted": false,
"conditionallySuccessful": false
}
]
},
{
"lhs": "8",
"rhs": "2 * 4",
"testExpression": "(8)-(2 * 4)",
"testCalculations": [
{
"result": "SUCCESS",
"testTitle": "Simple",
"testExpression": "simplify((8)-(2 * 4))",
"resultExpression": "0",
"wasAborted": false,
"conditionallySuccessful": false
}
]
}
]
}
}
},
"positions": [
{
"section": 3,
"sentence": 1,
"word": 25
}
],
"includes": [],
"isPartOf": [],
"definiens": []
}
]
}
|
| 43 | =2
|
Gegenbauer polynomials | Full data:
{
"id": 43,
"pid": 93,
"eid": "math.93.0-1",
"title": "Gegenbauer polynomials",
"formulae": [
{
"id": "FORMULA_34d9d355f0c0e28d91465c3b575fb0a1",
"formula": "=2",
"semanticFormula": "\\alpha = 2",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "\\[Alpha] = 2"
},
"Maple": {
"translation": "alpha = 2"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 17
}
],
"includes": [],
"isPartOf": [
"\\begin{align}C_0^\\alpha(x) & = 1 \\\\C_1^\\alpha(x) & = 2 \\alpha x \\\\C_n^\\alpha(x) & = \\frac{1}{n}[2x(n+\\alpha-1)C_{n-1}^\\alpha(x) - (n+2\\alpha-2)C_{n-2}^\\alpha(x)].\\end{align}"
],
"definiens": [
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| 44 | \lim_{q\to 1}\;_{j}\phi_k \left[\begin{matrix} q^{a_1} & q^{a_2} & \ldots & q^{a_j} \\ q^{b_1} & q^{b_2} & \ldots & q^{b_k} \end{matrix} ; q,(q-1)^{1+k-j} z \right]=\;_{j}F_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_j \\ b_1 & b_2 & \ldots & b_k \end{matrix} ;z \right]
|
Basic hypergeometric series | Full data:
{
"id": 44,
"pid": 94,
"eid": "math.94.4",
"title": "Basic hypergeometric series",
"formulae": [
{
"id": "FORMULA_33e3b57bb75d5ea3b5b8ddcceef38430",
"formula": "\\lim_{q\\to 1}\\;_{j}\\phi_k \\left[\\begin{matrix} q^{a_1} & q^{a_2} & \\ldots & q^{a_j} \\\\ q^{b_1} & q^{b_2} & \\ldots & q^{b_k} \\end{matrix} ; q,(q-1)^{1+k-j} z \\right]=\\;_{j}F_k \\left[\\begin{matrix} a_1 & a_2 & \\ldots & a_j \\\\ b_1 & b_2 & \\ldots & b_k \\end{matrix} ;z \\right]",
"semanticFormula": "\\lim_{q\\to 1} \\qgenhyperphi{j}{k}@{q^{a_1} , q^{a_2} , \\ldots , q^{a_j}}{q^{b_1} , q^{b_2} , \\ldots , q^{b_k}}{q}{(q - 1)^{1+k-j} z} = \\genhyperF{j}{k}@{a_1 , a_2 , \\ldots , a_j}{b_1 , b_2 , \\ldots , b_k}{z}",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 5,
"word": 13
}
],
"includes": [
"q^{n}",
"q",
"b",
"a",
"z"
],
"isPartOf": [],
"definiens": [
{
"definition": "q-analog of the hypergeometric series",
"score": 2
},
{
"definition": "unilateral basic hypergeometric series",
"score": 2
},
{
"definition": "basic hypergeometric series",
"score": 2
}
]
}
]
}
|
| 45 | \frac{d^2w}{dz^2}+\left(-\frac{1}{4}+\frac{\kappa}{z}+\frac{1/4-\mu^2}{z^2}\right)w=0
|
Whittaker function | Full data:
{
"id": 45,
"pid": 95,
"eid": "math.95.0",
"title": "Whittaker function",
"formulae": [
{
"id": "FORMULA_16ec3a3583ee2b4621d316bf839c1725",
"formula": "\\frac{d^2w}{dz^2}+\\left(-\\frac{1}{4}+\\frac{\\kappa}{z}+\\frac{1\/4-\\mu^2}{z^2}\\right)w=0",
"semanticFormula": "\\deriv [2]{w}{z} +(- \\frac{1}{4} + \\frac{\\kappa}{z} + \\frac{1\/4-\\mu^2}{z^2}) w = 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "D[w, {z, 2}]+(-Divide[1,4]+Divide[\\[Kappa],z]+Divide[1\/4 - \\[Mu]^(2),(z)^(2)])*w == 0"
},
"Maple": {
"translation": "diff(w, [z$(2)])+(-(1)\/(4)+(kappa)\/(z)+(1\/4 - (mu)^(2))\/((z)^(2)))*w = 0"
}
},
"positions": [
{
"section": 0,
"sentence": 2,
"word": 4
}
],
"includes": [
"\\mu",
"\\kappa",
"z"
],
"isPartOf": [],
"definiens": [
{
"definition": "Whittaker 's equation",
"score": 2
},
{
"definition": "Whittaker function",
"score": 1
}
]
}
]
}
|
| 46 | e_1=\tfrac12,\qquad e_2=0,\qquad e_3=-\tfrac12
|
Lemniscatic elliptic function | Full data:
{
"id": 46,
"pid": 96,
"eid": "math.96.1",
"title": "Lemniscatic elliptic function",
"formulae": [
{
"id": "FORMULA_24137d79f0a282f42fdf9ea93576e998",
"formula": "e_1=\\tfrac12,\\qquad e_2=0,\\qquad e_3=-\\tfrac12",
"semanticFormula": "e_1=\\tfrac12,\\qquad e_2=0,\\qquad e_3=-\\tfrac12",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Subscript[e, 1] == Divide[1,2]\n Subscript[e, 2] = 0\n Subscript[e, 3] = -Divide[1,2]"
},
"Maple": {
"translation": "e[1] := (1)\/(2); e[2] := 0; e[3] := -(1)\/(2)"
}
},
"positions": [
{
"section": 0,
"sentence": 5,
"word": 11
}
],
"includes": [
"e_{1}",
"e_{2}",
"e_{3}"
],
"isPartOf": [],
"definiens": [
{
"definition": "constant",
"score": 2
}
]
}
]
}
|
| 47 | \gamma> 0,n-p=m-q> 0
|
Meijer G-function | Full data:
{
"id": 47,
"pid": 98,
"eid": "math.98.53-1",
"title": "Meijer G-function",
"formulae": [
{
"id": "FORMULA_028eb01ef675c90ea0f74fcdd93fc78c",
"formula": "\\gamma> 0,n-p=m-q> 0",
"semanticFormula": "\\gamma> 0,n-p=m-q> 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "\\[Gamma] > 0\n n - p == m - q > 0"
},
"Maple": {
"translation": "gamma > 0; n - p = m - q > 0"
}
},
"positions": [
{
"section": 12,
"sentence": 0,
"word": 17
}
],
"includes": [
"m",
"q",
"p=q> 0",
"n",
"p=q",
"\\gamma>"
],
"isPartOf": [],
"definiens": [
{
"definition": "constraint",
"score": 2
}
]
}
]
}
|
| 48 | \begin{pmatrix} j \\ m \quad m'\end{pmatrix}:= \sqrt{2 j + 1}\begin{pmatrix} j & 0 & j \\ m & 0 & m'\end{pmatrix}= (-1)^{j - m'} \delta_{m, -m'}
|
3-j symbol | Full data:
{
"id": 48,
"pid": 99,
"eid": "math.99.30",
"title": "3-j symbol",
"formulae": [
{
"id": "FORMULA_3f987b881a59a03904ff9a79476faae0",
"formula": "\\begin{pmatrix} j \\\\ m \\quad m'\\end{pmatrix}:= \\sqrt{2 j + 1}\\begin{pmatrix} j & 0 & j \\\\ m & 0 & m'\\end{pmatrix}= (-1)^{j - m'} \\delta_{m, -m'}",
"semanticFormula": "\\begin{pmatrix} j \\\\ m \\quad m'\\end{pmatrix}:= \\sqrt{2 j + 1}\\begin{pmatrix} j & 0 & j \\\\ m & 0 & m'\\end{pmatrix}= (-1)^{j - m'} \\delta_{m, -m'}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Wigner[j_, m_, m\\[Prime]_] := Sqrt[2*j+1] * {{j, 0, j}, {m, 0, m\\[Prime]}} = (-1)^(j-m\\[Prime])*Subscript[\\[Delta], m, -m\\[Prime]]"
}
},
"positions": [
{
"section": 10,
"sentence": 0,
"word": 23
}
],
"includes": [
"j",
"m"
],
"isPartOf": [],
"definiens": [
{
"definition": "Wigner 1-jm symbol",
"score": 2
},
{
"definition": "metric tensor in angular-momentum theory",
"score": 2
},
{
"definition": "quantity",
"score": 0
}
]
}
]
}
|
| 49 | \begin{Bmatrix} i & j & \ell\\ k & m & n \end{Bmatrix}= (\Phi_{i,j}^{k,m})_{\ell,n}
|
6-j symbol | Full data:
{
"id": 49,
"pid": 100,
"eid": "math.100.14",
"title": "6-j symbol",
"formulae": [
{
"id": "FORMULA_21d6ec52b25bb130bf068c4857bbcb93",
"formula": "\\begin{Bmatrix} i & j & \\ell\\\\ k & m & n \\end{Bmatrix}= (\\Phi_{i,j}^{k,m})_{\\ell,n}",
"semanticFormula": "\\Wignersixjsym{i}{j}{\\ell}{k}{m}{n} = (\\Phi_{i,j}^{k,m})_{\\ell,n}",
"confidence": 0.8624533614429312,
"translations": {},
"positions": [
{
"section": 5,
"sentence": 4,
"word": 23
}
],
"includes": [
"j"
],
"isPartOf": [],
"definiens": [
{
"definition": "6j symbol",
"score": 2
},
{
"definition": "associativity isomorphism",
"score": 2
},
{
"definition": "symbol",
"score": 1
},
{
"definition": "vector space isomorphism",
"score": 2
},
{
"definition": "Wigner",
"score": 1
},
{
"definition": "Wigner 's 6 - j symbol",
"score": 2
}
]
}
]
}
|
| 50 | \sum_{j_7 j_8} (2j_7+1)(2j_8+1) \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} \begin{Bmatrix} j_1 & j_2 & j_3'\\ j_4 & j_5 & j_6'\\ j_7 & j_8 & j_9 \end{Bmatrix} = \frac{\delta_{j_3j_3'}\delta_{j_6j_6'} \begin{Bmatrix} j_1 & j_2 & j_3 \end{Bmatrix} \begin{Bmatrix} j_4 & j_5 & j_6\end{Bmatrix} \begin{Bmatrix} j_3 & j_6 & j_9 \end{Bmatrix}} {(2j_3+1)(2j_6+1)}
|
9-j symbol | Full data:
{
"id": 50,
"pid": 101,
"eid": "math.101.32",
"title": "9-j symbol",
"formulae": [
{
"id": "FORMULA_08d08037d9e64d85aa3645470ce645af",
"formula": "\\sum_{j_7 j_8} (2j_7+1)(2j_8+1) \\begin{Bmatrix} j_1 & j_2 & j_3\\\\ j_4 & j_5 & j_6\\\\ j_7 & j_8 & j_9 \\end{Bmatrix} \\begin{Bmatrix} j_1 & j_2 & j_3'\\\\ j_4 & j_5 & j_6'\\\\ j_7 & j_8 & j_9 \\end{Bmatrix} = \\frac{\\delta_{j_3j_3'}\\delta_{j_6j_6'} \\begin{Bmatrix} j_1 & j_2 & j_3 \\end{Bmatrix} \\begin{Bmatrix} j_4 & j_5 & j_6\\end{Bmatrix} \\begin{Bmatrix} j_3 & j_6 & j_9 \\end{Bmatrix}} {(2j_3+1)(2j_6+1)}",
"semanticFormula": "\\sum_{j_7 j_8} (2j_7+1)(2j_8+1) \\Wignerninejsym{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}{j_7}{j_8}{j_9} \\Wignerninejsym{j_1}{j_2}{j_3'}{j_4}{j_5}{j_6'}{j_7}{j_8}{j_9} = \\frac{\\delta_{j_3j_3'}\\delta_{j_6j_6'} \\begin{Bmatrix} j_1 & j_2 & j_3 \\end{Bmatrix} \\begin{Bmatrix} j_4 & j_5 & j_6\\end{Bmatrix} \\begin{Bmatrix} j_3 & j_6 & j_9 \\end{Bmatrix}}{(2j_3+1)(2j_6+1)}",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 5,
"sentence": 0,
"word": 10
}
],
"includes": [
"j",
"_{4}"
],
"isPartOf": [],
"definiens": [
{
"definition": "orthogonality relation",
"score": 1
},
{
"definition": "triangular delta",
"score": 2
},
{
"definition": "symbol",
"score": 1
},
{
"definition": "Wigner 's 9 - j symbol",
"score": 2
}
]
}
]
}
|
| 51 | \mathcal{K}_k(x; n,q) = \sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \binom {n-j}{k-j} \binom{x}{j}
|
Kravchuk polynomials | Full data:
{
"id": 51,
"pid": 102,
"eid": "math.102.5",
"title": "Kravchuk polynomials",
"formulae": [
{
"id": "FORMULA_6b7eb62a3e02e45fb1365dd2f07a5bbc",
"formula": "\\mathcal{K}_k(x; n,q) = \\sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \\binom {n-j}{k-j} \\binom{x}{j}",
"semanticFormula": "\\KrawtchoukpolyK{k}@{x}{n}{q} = \\sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \\binom {n-j}{k-j} \\binom{x}{j}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "K[k_, x_, n_, q_] := Sum[(- q)^(j)*(q - 1)^(k - j)*Binomial[n - j,k - j]*Binomial[x,j], {j, 0, k}, GenerateConditions->None]"
}
},
"positions": [
{
"section": 2,
"sentence": 0,
"word": 9
}
],
"includes": [
"q",
"n"
],
"isPartOf": [],
"definiens": [
{
"definition": "following alternative expression",
"score": 0
},
{
"definition": "Kravchuk polynomial",
"score": 2
}
]
}
]
} |
| 52 | g_1(x) = \sum_{k \geq 1} \frac{\sin(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2
|
Kelvin functions | Full data:
{
"id": 52,
"pid": 103,
"eid": "math.103.8",
"title": "Kelvin functions",
"formulae": [
{
"id": "FORMULA_07453e6baf8f216467f9b664de795bfc",
"formula": "g_1(x) = \\sum_{k \\geq 1} \\frac{\\sin(k \\pi \/ 4)}{k! (8x)^k} \\prod_{l = 1}^k (2l - 1)^2",
"semanticFormula": "g_1(x) = \\sum_{k \\geq 1} \\frac{\\sin(k \\cpi \/ 4)}{k! (8x)^k} \\prod_{l = 1}^k(2 l - 1)^2",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Subscript[g, 1][x_] := Sum[Divide[Sin[k*Pi\/4],(k)!*(8*x)^(k)]*Product[(2*l - 1)^(2), {l, 1, k}, GenerateConditions->None], {k, 1, Infinity}, GenerateConditions->None]"
},
"Maple": {
"translation": "g[1] := (x) -> sum((sin(k*Pi\/4))\/(factorial(k)*(8*x)^(k))*product((2*l - 1)^(2), l = 1..k), k = 1..infinity)"
}
},
"positions": [
{
"section": 1,
"sentence": 1,
"word": 28
}
],
"includes": [
"x",
"x)",
"g_1(x)"
],
"isPartOf": [],
"definiens": [
{
"definition": "series expansion",
"score": 1
},
{
"definition": "special case",
"score": 0
},
{
"definition": "asymptotic series",
"score": 1
},
{
"definition": "definition",
"score": 2
}
]
}
]
} |
| 53 | S_{\mu,\nu}(z) = s_{\mu,\nu}(z) + 2^{\mu-1} \Gamma\left(\frac{\mu + \nu + 1}{2}\right) \Gamma\left(\frac{\mu - \nu + 1}{2}\right)\left(\sin \left[(\mu - \nu)\frac{\pi}{2}\right] J_\nu(z) - \cos \left[(\mu - \nu)\frac{\pi}{2}\right] Y_\nu(z)\right)
|
Lommel function | Full data:
{
"id": 53,
"pid": 104,
"eid": "math.104.2",
"title": "Lommel function",
"formulae": [
{
"id": "FORMULA_03f5cb50caaedb9f0a4ada231fd61c58",
"formula": "S_{\\mu,\\nu}(z) = s_{\\mu,\\nu}(z) + 2^{\\mu-1} \\Gamma\\left(\\frac{\\mu + \\nu + 1}{2}\\right) \\Gamma\\left(\\frac{\\mu - \\nu + 1}{2}\\right)\\left(\\sin \\left[(\\mu - \\nu)\\frac{\\pi}{2}\\right] J_\\nu(z) - \\cos \\left[(\\mu - \\nu)\\frac{\\pi}{2}\\right] Y_\\nu(z)\\right)",
"semanticFormula": "\\LommelS{\\mu}{\\nu}@{z} = \\Lommels{\\mu}{\\nu}@{z} + 2^{\\mu-1} \\EulerGamma@{\\frac{\\mu + \\nu + 1}{2}} \\EulerGamma@{\\frac{\\mu - \\nu + 1}{2}}(\\sin [(\\mu - \\nu) \\frac{\\cpi}{2}] \\BesselJ{\\nu}@{z} - \\cos [(\\mu - \\nu) \\frac{\\cpi}{2}] \\BesselY{\\nu}@{z})",
"confidence": 0.8775479393290169,
"translations": {
"Mathematica": {
"translation": "S[\\[Mu]_, \\[Nu]_, z_] := Divide[Pi,2]*(BesselY[\\[Nu], z]*Integrate[(x)^\\[Mu]* BesselJ[\\[Nu], x], {x, 0, z}, GenerateConditions->None]- BesselJ[\\[Nu], z]*Integrate[(x)^\\[Mu]* BesselY[\\[Nu], x], {x, 0, z}, GenerateConditions->None]) + (2)^(\\[Mu]- 1)* Gamma[Divide[\\[Mu]+ \\[Nu]+ 1,2]]*Gamma[Divide[\\[Mu]- \\[Nu]+ 1,2]]*(Sin[((\\[Mu]- \\[Nu])*Divide[Pi,2])*]*BesselJ[\\[Nu], z]- Cos[((\\[Mu]- \\[Nu])*Divide[Pi,2])*]*BesselY[\\[Nu], z])"
},
"Maple": {
"translation": "LommelS1(mu, nu, z) = (Pi)\/(2)*(BesselY(nu, z)*int((x)^(mu)* BesselJ(nu, x), x = 0..z)- BesselJ(nu, z)*int((x)^(mu)* BesselY(nu, x), x = 0..z))"
}
},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 18
}
],
"includes": [
"s_{\\mu,\\nu}(z)",
"S_{\\mu,\\nu}(z)",
"J_{\\nu}(z)",
"Y_{\\nu}(z)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Lommel function",
"score": 2
},
{
"definition": "Bessel function of the first kind",
"score": 2
},
{
"definition": "Bessel function of the second kind",
"score": 2
}
]
}
]
} |
| 54 | \mathbf{H}_{\alpha}(z) = \frac{z^{\alpha+1}}{2^{\alpha}\sqrt{\pi} \Gamma \left (\alpha+\tfrac{3}{2} \right )} {}_1F_2 \left (1,\tfrac{3}{2}, \alpha+\tfrac{3}{2},-\tfrac{z^2}{4} \right )
|
Struve function | Full data:
{
"id": 54,
"pid": 105,
"eid": "math.105.18",
"title": "Struve function",
"formulae": [
{
"id": "FORMULA_6dc2da7f595d2f199fbc15768167f006",
"formula": "\\mathbf{H}_{\\alpha}(z) = \\frac{z^{\\alpha+1}}{2^{\\alpha}\\sqrt{\\pi} \\Gamma \\left (\\alpha+\\tfrac{3}{2} \\right )} {}_1F_2 \\left (1,\\tfrac{3}{2}, \\alpha+\\tfrac{3}{2},-\\tfrac{z^2}{4} \\right )",
"semanticFormula": "\\StruveH{\\alpha}@{z} = \\frac{z^{\\alpha+1}}{2^{\\alpha} \\sqrt{\\cpi} \\EulerGamma@{\\alpha + \\tfrac{3}{2}}} \\genhyperF{1}{2}@{1}{\\tfrac{3}{2}, \\alpha + \\tfrac{3}{2}}{- \\tfrac{z^2}{4}}",
"confidence": 0.8740850655136605,
"translations": {
"Mathematica": {
"translation": "StruveH[\\[Alpha], z] == Divide[(z)^(\\[Alpha]+ 1),(2)^\\[Alpha]*Sqrt[Pi]*Gamma[\\[Alpha]+Divide[3,2]]]*HypergeometricPFQ[{1}, {Divide[3,2], \\[Alpha]+Divide[3,2]}, -Divide[(z)^(2),4]]"
},
"Maple": {
"translation": "StruveH(alpha, z) = ((z)^(alpha + 1))\/((2)^(alpha)*sqrt(Pi)*GAMMA(alpha +(3)\/(2)))*hypergeom([1], [(3)\/(2), alpha +(3)\/(2)], -((z)^(2))\/(4))"
}
},
"positions": [
{
"section": 6,
"sentence": 2,
"word": 31
}
],
"includes": [
"_{1}F_{2}",
"\\mathbf{K}_\\alpha(x)",
"\\alpha",
"\\Gamma(z)",
"\\mathbf{H}_{\\alpha}(x)",
"\\mathbf{L}_{\\alpha}(x)",
"\\mathbf{H}_{\\alpha}(z)",
"Y_{\\alpha}(x)",
"\\mathbf{M}_\\alpha(x)"
],
"isPartOf": [],
"definiens": [
{
"definition": "hypergeometric function",
"score": 2
},
{
"definition": "Struve",
"score": 2
},
{
"definition": "Struve function",
"score": 2
},
{
"definition": "gamma function",
"score": 2
}
]
}
]
} |
| 55 | f(t+p) = f(t)
|
Hill differential equation | Full data:
{
"id": 55,
"pid": 106,
"eid": "math.106.7",
"title": "Hill differential equation",
"formulae": [
{
"id": "FORMULA_3a6745862e8f6ef2b93c343ad82b40c0",
"formula": "f(t+p) = f(t)",
"semanticFormula": "f(t+p) = f(t)",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "f[t + p] == f[t]"
},
"Maple": {
"translation": "f(t + p) = f(t)"
}
},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 21
}
],
"includes": [
"f(t)",
"t",
"p",
"f(t+\\pi)=f(t)"
],
"isPartOf": [
"f(t+\\pi)=f(t)"
],
"definiens": [
{
"definition": "function",
"score": 2
},
{
"definition": "periodic function by minimal period",
"score": 2
}
]
}
]
} |
| 56 | \mathbf{J}_\nu(z)=\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}+\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}
|
Anger function | Full data:
{
"id": 56,
"pid": 108,
"eid": "math.108.3",
"title": "Anger function",
"formulae": [
{
"id": "FORMULA_014efde25f995ccd08168a36ec7ef86d",
"formula": "\\mathbf{J}_\\nu(z)=\\cos\\frac{\\pi\\nu}{2}\\sum_{k=0}^\\infty\\frac{(-1)^kz^{2k}}{4^k\\Gamma\\left(k+\\frac{\\nu}{2}+1\\right)\\Gamma\\left(k-\\frac{\\nu}{2}+1\\right)}+\\sin\\frac{\\pi\\nu}{2}\\sum_{k=0}^\\infty\\frac{(-1)^kz^{2k+1}}{2^{2k+1}\\Gamma\\left(k+\\frac{\\nu}{2}+\\frac{3}{2}\\right)\\Gamma\\left(k-\\frac{\\nu}{2}+\\frac{3}{2}\\right)}",
"semanticFormula": "\\AngerJ{\\nu}@{z} = \\cos \\frac{\\cpi\\nu}{2} \\sum_{k=0}^\\infty \\frac{(-1)^k z^{2k}}{4^k\\EulerGamma@{k+\\frac{\\nu}{2}+1}\\EulerGamma@{k-\\frac{\\nu}{2}+1}}+\\sin\\frac{\\cpi\\nu}{2}\\sum_{k=0}^\\infty\\frac{(-1)^k z^{2k+1}}{2^{2k+1}\\EulerGamma@{k+\\frac{\\nu}{2}+\\frac{3}{2}}\\EulerGamma@{k-\\frac{\\nu}{2}+\\frac{3}{2}}}",
"confidence": 0.8648813564530858,
"translations": {
"Mathematica": {
"translation": "AngerJ[\\[Nu], z] == Cos[Divide[Pi*\\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k),(4)^(k)* Gamma[k +Divide[\\[Nu],2]+ 1]*Gamma[k -Divide[\\[Nu],2]+ 1]], {k, 0, Infinity}, GenerateConditions->None]+ Sin[Divide[Pi*\\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k + 1),(2)^(2*k + 1)* Gamma[k +Divide[\\[Nu],2]+Divide[3,2]]*Gamma[k -Divide[\\[Nu],2]+Divide[3,2]]], {k, 0, Infinity}, GenerateConditions->None]"
},
"Maple": {
"translation": "AngerJ(nu, z) = cos((Pi*nu)\/(2))*sum(((- 1)^(k)* (z)^(2*k))\/((4)^(k)* GAMMA(k +(nu)\/(2)+ 1)*GAMMA(k -(nu)\/(2)+ 1)), k = 0..infinity)+ sin((Pi*nu)\/(2))*sum(((- 1)^(k)* (z)^(2*k + 1))\/((2)^(2*k + 1)* GAMMA(k +(nu)\/(2)+(3)\/(2))*GAMMA(k -(nu)\/(2)+(3)\/(2))), k = 0..infinity)"
}
},
"positions": [
{
"section": 2,
"sentence": 0,
"word": 8
}
],
"includes": [
"J_{\\nu}",
"\\mathbf{J}_{\\nu}",
"\\nu"
],
"isPartOf": [],
"definiens": [
{
"definition": "power series expansion",
"score": 2
},
{
"definition": "Anger function",
"score": 2
},
{
"definition": "Gamma function",
"score": 2
}
]
}
]
} |
| 57 | (\operatorname{Ec})^'_{2K} = (\operatorname{Ec})^'_0 = 0, \;\; (\operatorname{Es})^'_{2K} = (\operatorname{Es})^'_0 = 0
|
Lamé function | Full data:
{
"id": 57,
"pid": 109,
"eid": "math.109.27",
"title": "Lam\u00e9 function",
"formulae": [
{
"id": "FORMULA_7d20395e75eeb74df48a681897d9d727",
"formula": "(\\operatorname{Ec})^'_{2K} = (\\operatorname{Ec})^'_0 = 0, \\;\\; (\\operatorname{Es})^'_{2K} = (\\operatorname{Es})^'_0 = 0",
"semanticFormula": "(\\operatorname{Ec})_{2K}^' =(\\operatorname{Ec})_0^' = 0 ,(\\operatorname{Es})_{2K}^' =(\\operatorname{Es})_0^' = 0",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 2,
"sentence": 3,
"word": 31
}
],
"includes": [
"\\operatorname{Ec}",
"\\operatorname{Es}"
],
"isPartOf": [],
"definiens": [
{
"definition": "boundary condition",
"score": 2
},
{
"definition": "ellipsoidal wave",
"score": 2
}
]
}
]
} |
| 58 | \int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)
|
Gauss–Hermite quadrature | Full data:
{
"id": 58,
"pid": 110,
"eid": "math.110.1",
"title": "Gauss\u2013Hermite quadrature",
"formulae": [
{
"id": "FORMULA_cdf8d887d4b5ad1a7724773d8eef8fd2",
"formula": "\\int_{-\\infty}^{+\\infty} e^{-x^2} f(x)\\,dx \\approx \\sum_{i=1}^n w_i f(x_i)",
"semanticFormula": "\\int_{-\\infty}^{+\\infty} \\expe^{-x^2} f(x) \\diff{x} \\approx \\sum_{i=1}^n w_i f(x_i)",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 3
}
],
"includes": [
"\\int_{-\\infty}^{+\\infty} e^{-x^2} f(x)\\,dx",
"n",
"x_{i}",
"w_{i}"
],
"isPartOf": [],
"definiens": [
{
"definition": "value of integral",
"score": 2
},
{
"definition": "form of Gaussian quadrature",
"score": 2
},
{
"definition": "Gauss -- Hermite quadrature",
"score": 2
},
{
"definition": "Hermite polynomial",
"score": 1
},
{
"definition": "associated weight",
"score": 2
}
]
}
]
} |
| 59 | p_n(x;a,b,c,d|q) =(ab,ac,ad;q)_na^{-n}\;_{4}\phi_3 \left[\begin{matrix} q^{-n}&abcdq^{n-1}&ae^{i\theta}&ae^{-i\theta} \\ ab&ac&ad \end{matrix} ; q,q \right]
|
Askey–Wilson polynomials | Full data:
{
"id": 59,
"pid": 111,
"eid": "math.111.0",
"title": "Askey\u2013Wilson polynomials",
"formulae": [
{
"id": "FORMULA_cfe946a0547913234ac79d398f269607",
"formula": "p_n(x;a,b,c,d|q) =(ab,ac,ad;q)_na^{-n}\\;_{4}\\phi_3 \\left[\\begin{matrix} q^{-n}&abcdq^{n-1}&ae^{i\\theta}&ae^{-i\\theta} \\\\ ab&ac&ad \\end{matrix} ; q,q \\right]",
"semanticFormula": "\\AskeyWilsonpolyp{n}@{x}{a}{b}{c}{d}{q} = \\qmultiPochhammersym{ab , ac , ad}{q}{n} a^{-n} \\qgenhyperphi{4}{3}@{q^{-n} , abcdq^{n-1} , a\\expe^{\\iunit\\theta} , a\\expe^{-\\iunit\\theta}}{ab , ac , ad}{q}{q}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "p[n_, x_, a_, b_, c_, d_, q_] := Product[QPochhammer[Part[{a*b , a*c , a*d},i],q,n],{i,1,Length[{a*b , a*c , a*d}]}]*(a)^(- n)* QHypergeometricPFQ[{(q)^(- n), a*b*c*d*(q)^(n - 1), a*Exp[I*\\[Theta]], a*Exp[- I*\\[Theta]]},{a*b , a*c , a*d},q,q]"
}
},
"positions": [
{
"section": 0,
"sentence": 3,
"word": 4
}
],
"includes": [
"\\phi",
"_{n}",
"n"
],
"isPartOf": [],
"definiens": [
{
"definition": "basic hypergeometric function",
"score": 2
},
{
"definition": "q-Pochhammer symbol",
"score": 2
},
{
"definition": "Askey\u2013Wilson polynomials",
"score": 2
}
]
}
]
} |
| 60 | Q_n(x;\alpha,\beta,N)= {}_3F_2(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1).
|
Hahn polynomials | Full data:
{
"id": 60,
"pid": 112,
"eid": "math.112.0",
"title": "Hahn polynomials",
"formulae": [
{
"id": "FORMULA_777007203448847310455e0b0eaaeb2c",
"formula": "Q_n(x;\\alpha,\\beta,N)= {}_3F_2(-n,-x,n+\\alpha+\\beta+1;\\alpha+1,-N+1;1).",
"semanticFormula": "\\HahnpolyQ{n}@{x}{\\alpha}{\\beta}{N} = \\genhyperF{3}{2}@{- n , - x , n + \\alpha + \\beta + 1}{\\alpha + 1 , - N + 1}{1}",
"confidence": 0.8953028732079359,
"translations": {
"Mathematica": {
"translation": "Q[n_, x_, \\[Alpha]_, \\[Beta]_, N_] := HypergeometricPFQ[{- n , - x , n + \\[Alpha]+ \\[Beta]+ 1}, {\\[Alpha]+ 1 , - N + 1}, 1]"
}
},
"positions": [
{
"section": 0,
"sentence": 3,
"word": 11
}
],
"includes": [
"R_{n}(x;\\gamma,\\delta,N)",
"S_{n}(x;a,b,c)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Hahn polynomial",
"score": 2
},
{
"definition": "basic hypergeometric function",
"score": 2
},
{
"definition": "hypergeometric function",
"score": 2
}
]
}
]
} |
| 61 | \sum_{x=0}^\infty \frac{\mu^x}{x!} C_n(x; \mu)C_m(x; \mu)=\mu^{-n} e^\mu n! \delta_{nm}, \quad \mu>0
|
Charlier polynomials | Full data:
{
"id": 61,
"pid": 113,
"eid": "math.113.2",
"title": "Charlier polynomials",
"formulae": [
{
"id": "FORMULA_b76bcf7237b989f6b5d90082fafa53f1",
"formula": "\\sum_{x=0}^\\infty \\frac{\\mu^x}{x!} C_n(x; \\mu)C_m(x; \\mu)=\\mu^{-n} e^\\mu n! \\delta_{nm}, \\quad \\mu>0",
"semanticFormula": "\\sum_{x=0}^\\infty \\frac{\\mu^x}{x!} \\CharlierpolyC{n}@{x}{\\mu} \\CharlierpolyC{m}@{x}{\\mu} = \\mu^{-n} \\expe^\\mu n! \\delta_{nm} , \\quad \\mu > 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Sum[Divide[\\[Mu]^x, x!] * HypergeometricPFQ[{-n, -x}, {}, -Divide[1,\\[Mu]]] * HypergeometricPFQ[{-m, -x}, {}, -Divide[1,\\[Mu]]], {x, 0, Infinity}] == \\[Mu]^(-n)*Exp[\\[Mu]]*n!*Subscript[\\[Delta], n, m]"
}
},
"positions": [
{
"section": 0,
"sentence": 2,
"word": 5
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "orthogonality relation",
"score": 2
},
{
"definition": "Charlier polynomial",
"score": 2
}
]
}
]
} |
| 62 | p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q) = {}_4\phi_3\left[\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&q^{x+1}cd\\aq&bdq&cq\\ \end{matrix};q;q\right]
|
Q-Racah polynomials | Full data:
{
"id": 62,
"pid": 114,
"eid": "math.114.0",
"title": "Q-Racah polynomials",
"formulae": [
{
"id": "FORMULA_51c23bddc19530680328afbf28235b90",
"formula": "p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q) = {}_4\\phi_3\\left[\\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&q^{x+1}cd\\\\aq&bdq&cq\\\\ \\end{matrix};q;q\\right]",
"semanticFormula": "\\qRacahpolyR{n}@{q^{-x} + q^{x+1} cd}{a}{b}{c}{d}{q} = \\qgenhyperphi{4}{3}@{q^{-n}, abq^{n+1}, q^{-x}, q^{x+1}cd}{aq , bdq , cq}{q}{q}",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "term of basic hypergeometric function",
"score": 2
}
]
}
]
} |
| 63 | \displaystyle c_n(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a)
|
Q-Charlier polynomials | Full data:
{
"id": 63,
"pid": 115,
"eid": "math.115.0",
"title": "Q-Charlier polynomials",
"formulae": [
{
"id": "FORMULA_925d68ff3ddf733a69ec9936dfede5d6",
"formula": "\\displaystyle c_n(q^{-x};a;q) = {}_2\\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}\/a)",
"semanticFormula": "c_n(q^{-x} ; a ; q) = \\qgenhyperphi{2}{1}@{q^{-n} , q^{-x}}{0}{q}{- q^{n+1} \/ a}",
"confidence": 0.5776294951318733,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "q-Charlier polynomial",
"score": 2
},
{
"definition": "term of the basic hypergeometric function",
"score": 2
}
]
}
]
} |
| 64 | M_n(x,\beta,\gamma) = \sum_{k=0}^n (-1)^k{n \choose k}{x\choose k}k!(x+\beta)_{n-k}\gamma^{-k}
|
Meixner polynomials | Full data:
{
"id": 64,
"pid": 116,
"eid": "math.116.0",
"title": "Meixner polynomials",
"formulae": [
{
"id": "FORMULA_29a1f82de004c5721c8dfc5dd1dc5b98",
"formula": "M_n(x,\\beta,\\gamma) = \\sum_{k=0}^n (-1)^k{n \\choose k}{x\\choose k}k!(x+\\beta)_{n-k}\\gamma^{-k}",
"semanticFormula": "\\MeixnerpolyM{n}@{x}{\\beta}{\\gamma} = \\sum_{k=0}^n(- 1)^k{n \\choose k}{x\\choose k} k! \\Pochhammersym{x + \\beta}{n-k} \\gamma^{-k}",
"confidence": 0.8953028732079359,
"translations": {
"Mathematica": {
"translation": "M[n_, x_, \\[Beta]_, \\[Gamma]_] := Sum[(- 1)^(k)*Binomial[n,k]*Binomial[x,k]*(k)!*Pochhammer[x + \\[Beta], n - k]*\\[Gamma]^(- k), {k, 0, n}, GenerateConditions->None]"
}
},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 16
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "Meixner polynomial",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "term of binomial coefficient",
"score": 1
}
]
}
]
} |
| 65 | x(1-x) \frac {\partial^2F_1(x,y)} {\partial x^2} + y(1-x) \frac {\partial^2F_1(x,y)} {\partial x \partial y} + [c - (a+b_1+1) x] \frac {\partial F_1(x,y)} {\partial x} - b_1 y \frac {\partial F_1(x,y)} {\partial y} - a b_1 F_1(x,y) = 0
|
Appell series | Full data:
{
"id": 65,
"pid": 117,
"eid": "math.117.19",
"title": "Appell series",
"formulae": [
{
"id": "FORMULA_85014aaf0c7c1f4fe433115e796a03db",
"formula": "x(1-x) \\frac {\\partial^2F_1(x,y)} {\\partial x^2} + y(1-x) \\frac {\\partial^2F_1(x,y)} {\\partial x \\partial y} + [c - (a+b_1+1) x] \\frac {\\partial F_1(x,y)} {\\partial x} - b_1 y \\frac {\\partial F_1(x,y)} {\\partial y} - a b_1 F_1(x,y) = 0",
"semanticFormula": "x(1-x) \\deriv[2]{\\AppellF{1}@{a}{b_1}{b_2}{\\gamma}{x}{y}}{x} + y(1-x) \\frac{\\pdiff[2]{\\AppellF{1}@{a}{b_1}{b_2}{\\gamma}{x}{y}}}{\\pdiff{x}\\pdiff{y}} + [c - (a+b_1+1) x] \\deriv[1]{\\AppellF{1}@{a}{b_1}{b_2}{\\gamma}{x}{y}}{x} - b_1 y \\deriv[1]{\\AppellF{1}@{a}{b_1}{b_2}{\\gamma}{x}{y}}{y} - a b_1 \\AppellF{1}@{a}{b_1}{b_2}{\\gamma}{x}{y} = 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "x*(1-x) * D[AppellF[a, Subscript[b, 1], Subscript[b, 2], \\[Gamma], x, y], {x,2}] + y*(1-x) * D[AppellF[a, Subscript[b, 1], Subscript[b, 2], \\[Gamma], x, y], x, y] + (c - (a+Subscript[b, 1]+1)*x) * D[AppellF[a, Subscript[b, 1], Subscript[b, 2], \\[Gamma], x, y], x] - Subscript[b,1] * y * D[AppellF[a, Subscript[b, 1], Subscript[b, 2], \\[Gamma], x, y], y] - a*Subscript[b,1]*AppellF[a, Subscript[b, 1], Subscript[b, 2], \\[Gamma], x, y] == 0"
}
},
"positions": [
{
"section": 3,
"sentence": 0,
"word": 39
}
],
"includes": [
"y",
"x",
"F_{1}",
"F",
"_{1}F_{1}"
],
"isPartOf": [],
"definiens": [
{
"definition": "Appell",
"score": 2
},
{
"definition": "partial differential equation",
"score": 2
},
{
"definition": "system of differential equation",
"score": 1
},
{
"definition": "system of second-order differential equation",
"score": 2
},
{
"definition": "Appell series",
"score": 2
}
]
}
]
} |
| 66 | \Theta_\Lambda(\tau) = \sum_{x\in\Lambda}e^{i\pi\tau\|x\|^2}\qquad\mathrm{Im}\,\tau > 0
|
Theta function of a lattice | Full data:
{
"id": 66,
"pid": 118,
"eid": "math.118.0",
"title": "Theta function of a lattice",
"formulae": [
{
"id": "FORMULA_39f4baaa3543f22706b6f7701518f3eb",
"formula": "\\Theta_\\Lambda(\\tau) = \\sum_{x\\in\\Lambda}e^{i\\pi\\tau\\|x\\|^2}\\qquad\\mathrm{Im}\\,\\tau > 0",
"semanticFormula": "\\Theta_\\Lambda(\\tau) = \\sum_{x\\in\\Lambda} \\expe^{\\iunit \\cpi \\tau \\|x \\|^2} \\qquad \\imagpart \\tau > 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "\\[CapitalTheta][\\[CapitalLambda]_, \\[Tau]_] := Sum[Exp[I*Pi*\\[Tau]*(Norm[x])^(2)], {x, \\[CapitalLambda]}]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"\\Lambda"
],
"isPartOf": [],
"definiens": [
{
"definition": "theta function",
"score": 2
},
{
"definition": "lattice",
"score": 1
},
{
"definition": "Theta function of a lattice",
"score": 1
}
]
}
]
} |
| 67 | \frac{d^2 S}{dz^2}+\left(\sum _{j=1}^N \frac{\gamma _j}{z - a_j} \right) \frac{dS}{dz} + \frac{V(z)}{\prod _{j=1}^N (z - a_j)}S = 0
|
Heine–Stieltjes polynomials | Full data:
{
"id": 67,
"pid": 119,
"eid": "math.119.0",
"title": "Heine\u2013Stieltjes polynomials",
"formulae": [
{
"id": "FORMULA_d673cd2334542e8f83f099798c4027b3",
"formula": "\\frac{d^2 S}{dz^2}+\\left(\\sum _{j=1}^N \\frac{\\gamma _j}{z - a_j} \\right) \\frac{dS}{dz} + \\frac{V(z)}{\\prod _{j=1}^N (z - a_j)}S = 0",
"semanticFormula": "\\deriv [2]{S}{z} +(\\sum_{j=1}^N \\frac{\\gamma _j}{z - a_j}) \\deriv[]{S}{z} + \\frac{V(z)}{\\prod _{j=1}^N (z - a_j)} S = 0",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 6
}
],
"includes": [
"V(z)",
"S",
"V"
],
"isPartOf": [],
"definiens": [
{
"definition": "form",
"score": 0
},
{
"definition": "Fuchsian equation",
"score": 2
},
{
"definition": "polynomial",
"score": 1
},
{
"definition": "degree",
"score": 0
},
{
"definition": "Edward Burr Van Vleck",
"score": 0
},
{
"definition": "Heine",
"score": 1
},
{
"definition": "polynomial solution",
"score": 1
},
{
"definition": "Stieltjes polynomial",
"score": 1
},
{
"definition": "Van Vleck polynomial",
"score": 1
}
]
}
]
} |
| 68 | w(x) = \frac{k}{\sqrt{\pi}} x^{-1/2} \exp(-k^2\log^2 x)
|
Stieltjes–Wigert polynomials | Full data:
{
"id": 68,
"pid": 120,
"eid": "math.120.0",
"title": "Stieltjes\u2013Wigert polynomials",
"formulae": [
{
"id": "FORMULA_583d3b9e00bbd73091b01f368d1a82c7",
"formula": "w(x) = \\frac{k}{\\sqrt{\\pi}} x^{-1\/2} \\exp(-k^2\\log^2 x)",
"semanticFormula": "w(x) = \\frac{k}{\\sqrt{\\cpi}} x^{-1\/2} \\exp(- k^2 \\log^2 x)",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "w[x_] := Divide[k,Sqrt[Pi]]*(x)^(- 1\/2)* Exp[- (k)^(2)* (Log[x])^(2)]"
},
"Maple": {
"translation": "w := (x) -> (k)\/(sqrt(Pi))*(x)^(- 1\/2)* exp(- (k)^(2)* (log(x))^(2))"
}
},
"positions": [
{
"section": 0,
"sentence": 0,
"word": 38
}
],
"includes": [
"\\frac{k}{\\sqrt{\\pi}} x^{-1\/2} \\exp \\left(-k^2 \\log^2 x \\right)"
],
"isPartOf": [],
"definiens": [
{
"definition": "weight function",
"score": 2
},
{
"definition": "positive real line",
"score": 0
},
{
"definition": "basic Askey scheme",
"score": 1
},
{
"definition": "family of basic hypergeometric orthogonal polynomial",
"score": 1
},
{
"definition": "mathematics",
"score": 0
},
{
"definition": "Stieltjes -- Wigert polynomial",
"score": 2
},
{
"definition": "Thomas Jan Stieltjes",
"score": 0
},
{
"definition": "Carl Severin Wigert",
"score": 0
},
{
"definition": "example of such weight function",
"score": 0
}
]
}
]
} |
| 69 | y^2=x(x-1)(x-\lambda)
|
Modular lambda function | Full data:
{
"id": 69,
"pid": 121,
"eid": "math.121.23",
"title": "Modular lambda function",
"formulae": [
{
"id": "FORMULA_4e5334aa6f5fa551b0718a2372816061",
"formula": "y^2=x(x-1)(x-\\lambda)",
"semanticFormula": "y^2=x(x-1)(x-\\lambda)",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "(y)^(2) == x*(x - 1)*(x - \\[Lambda])"
},
"Maple": {
"translation": "(y)^(2) = x*(x - 1)*(x - lambda)"
}
},
"positions": [
{
"section": 2,
"sentence": 5,
"word": 13
}
],
"includes": [
"\\lambda"
],
"isPartOf": [],
"definiens": [
{
"definition": "elliptic curve of Legendre form",
"score": 2
},
{
"definition": "relation to the j-invariant",
"score": 1
},
{
"definition": "relation to the j-invariant",
"score": 1
}
]
}
]
} |
| 70 | P_1^{(\lambda)}(x;\phi)=2(\lambda\cos\phi + x\sin\phi)
|
Meixner–Pollaczek polynomials | Full data:
{
"id": 70,
"pid": 122,
"eid": "math.122.3",
"title": "Meixner\u2013Pollaczek polynomials",
"formulae": [
{
"id": "FORMULA_96d19b4b504f801548c69064d662043b",
"formula": "P_1^{(\\lambda)}(x;\\phi)=2(\\lambda\\cos\\phi + x\\sin\\phi)",
"semanticFormula": "\\MeixnerPollaczekpolyP{\\lambda}{1}@{x}{\\phi} = 2(\\lambda \\cos \\phi + x \\sin \\phi)",
"confidence": 0.8953028732079359,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 9
}
],
"includes": [
"P_{m}^{(\\lambda)}(x;\\varphi)"
],
"isPartOf": [],
"definiens": [
{
"definition": "first few Meixner -- Pollaczek polynomial",
"score": 2
}
]
}
]
} |
| 71 | P_n^{(\alpha,\beta)}(z)=\frac{(\alpha+1)_n}{n!}\,{}_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\tfrac{1}{2}(1-z)\right)
|
Jacobi polynomials | Full data:
{
"id": 71,
"pid": 123,
"eid": "math.123.0",
"title": "Jacobi polynomials",
"formulae": [
{
"id": "FORMULA_c8b5b9184e45bca39744427c45693115",
"formula": "P_n^{(\\alpha,\\beta)}(z)=\\frac{(\\alpha+1)_n}{n!}\\,{}_2F_1\\left(-n,1+\\alpha+\\beta+n;\\alpha+1;\\tfrac{1}{2}(1-z)\\right)",
"semanticFormula": "\\JacobipolyP{\\alpha}{\\beta}{n}@{z} = \\frac{\\Pochhammersym{\\alpha + 1}{n}}{n!} \\genhyperF{2}{1}@{- n , 1 + \\alpha + \\beta + n}{\\alpha + 1}{\\tfrac{1}{2}(1 - z)}",
"confidence": 0.7595006538205181,
"translations": {
"Mathematica": {
"translation": "JacobiP[n, \\[Alpha], \\[Beta], z] == Divide[Pochhammer[\\[Alpha]+ 1, n],(n)!]*HypergeometricPFQ[{- n , 1 + \\[Alpha]+ \\[Beta]+ n}, {\\[Alpha]+ 1}, Divide[1,2]*(1 - z)]"
},
"Maple": {
"translation": "JacobiP(n, alpha, beta, z) = (pochhammer(alpha + 1, n))\/(factorial(n))*hypergeom([- n , 1 + alpha + beta + n], [alpha + 1], (1)\/(2)*(1 - z))"
}
},
"positions": [
{
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"sentence": 0,
"word": 12
}
],
"includes": [
"P_{n}^{(\\alpha, \\beta)}(x)",
"(\\alpha+1)_n",
"n",
"n + \\alpha + \\beta",
"P_{n}^{(\\alpha, \\beta)}",
"\\alpha,\\beta",
"z"
],
"isPartOf": [],
"definiens": [
{
"definition": "Pochhammer 's symbol",
"score": 2
},
{
"definition": "hypergeometric function",
"score": 2
},
{
"definition": "Jacobi polynomial",
"score": 2
}
]
}
]
} |
| 72 | S_n(x^2;a,b,c)= {}_3F_2(-n,a+ix,a-ix;a+b,a+c;1).
|
Continuous dual Hahn polynomials | Full data:
{
"id": 72,
"pid": 124,
"eid": "math.124.0",
"title": "Continuous dual Hahn polynomials",
"formulae": [
{
"id": "FORMULA_b0d448ba925dc6b2bf2ce32a1253dee4",
"formula": "S_n(x^2;a,b,c)= {}_3F_2(-n,a+ix,a-ix;a+b,a+c;1).",
"semanticFormula": "\\contdualHahnpolyS{n}@{x^2}{a}{b}{c} = \\genhyperF{3}{2}@{- n , a + \\iunit x , a - \\iunit x}{a + b , a + c}{1}",
"confidence": 0.7132263353695951,
"translations": {},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 10
}
],
"includes": [
"R_{n}(x;\\gamma,\\delta,N)"
],
"isPartOf": [],
"definiens": [
{
"definition": "hypergeometric function",
"score": 1
},
{
"definition": "dual Hahn polynomial",
"score": 1
},
{
"definition": "continuous Hahn polynomial",
"score": 1
},
{
"definition": "continuous dual Hahn polynomial",
"score": 2
}
]
}
]
} |
| 73 | P_n^{(\alpha,\beta)}=\lim_{t\to\infty}t^{-n}p_n\left(\tfrac12xt; \tfrac12(\alpha+1-it), \tfrac12(\beta+1+it), \tfrac12(\alpha+1+it), \tfrac12(\beta+1-it)\right)
|
Continuous Hahn polynomials | Full data:
{
"id": 73,
"pid": 125,
"eid": "math.125.15",
"title": "Continuous Hahn polynomials",
"formulae": [
{
"id": "FORMULA_ff971744100fef3b34b2c93b6adc3efb",
"formula": "P_n^{(\\alpha,\\beta)}=\\lim_{t\\to\\infty}t^{-n}p_n\\left(\\tfrac12xt; \\tfrac12(\\alpha+1-it), \\tfrac12(\\beta+1+it), \\tfrac12(\\alpha+1+it), \\tfrac12(\\beta+1-it)\\right)",
"semanticFormula": "\\JacobipolyP{\\alpha}{\\beta}{n}@{x} = \\lim_{t\\to\\infty} t^{-n} \\contHahnpolyp{n}@{\\tfrac12 xt}{\\tfrac12(\\alpha + 1 - \\iunit t)}{\\tfrac12(\\beta + 1 + \\iunit t)}{\\tfrac12(\\alpha + 1 + \\iunit t)}{\\tfrac12(\\beta + 1 - \\iunit t)}",
"confidence": 0.9041995034970904,
"translations": {
"Mathematica": {
"translation": "JacobiP[n, \\[Alpha], \\[Beta], x] == Limit[(t)^(- n)* I^(n)*Divide[Pochhammer[Divide[1,2]*(\\[Alpha]+ 1 - I*t) + Divide[1,2]*(\\[Alpha]+ 1 + I*t), n]*Pochhammer[Divide[1,2]*(\\[Alpha]+ 1 - I*t) + Divide[1,2]*(\\[Beta]+ 1 - I*t), n], (n)!] * HypergeometricPFQ[{-(n), n + 2*Re[Divide[1,2]*(\\[Alpha]+ 1 - I*t) + Divide[1,2]*(\\[Beta]+ 1 + I*t)] - 1, Divide[1,2]*(\\[Alpha]+ 1 - I*t) + I*(Divide[1,2]*x*t)}, {Divide[1,2]*(\\[Alpha]+ 1 - I*t) + Divide[1,2]*(\\[Alpha]+ 1 + I*t), Divide[1,2]*(\\[Alpha]+ 1 - I*t) + Divide[1,2]*(\\[Beta]+ 1 - I*t)}, 1], t -> Infinity, GenerateConditions->None]"
}
},
"positions": [
{
"section": 5,
"sentence": 2,
"word": 20
}
],
"includes": [
"p_{n}(x;a,b,c,d)",
"F_{n}",
"P_{n}^{(\\alpha,\\beta)}"
],
"isPartOf": [],
"definiens": [
{
"definition": "case of the continuous Hahn polynomial",
"score": 1
},
{
"definition": "Jacobi polynomial",
"score": 2
},
{
"definition": "continuous Hahn polynomial",
"score": 2
}
]
}
]
} |
| 74 | \sum^{b-1}_{s=a}w_n^{(c)}(s,a,b)w_m^{(c)}(s,a,b)\rho(s)[\Delta x(s-\frac{1}{2}) ]=\delta_{nm}d_n^2
|
Dual Hahn polynomials | Full data:
{
"id": 74,
"pid": 126,
"eid": "math.126.7",
"title": "Dual Hahn polynomials",
"formulae": [
{
"id": "FORMULA_657ec9a2e460e61adc6857260291be56",
"formula": "\\sum^{b-1}_{s=a}w_n^{(c)}(s,a,b)w_m^{(c)}(s,a,b)\\rho(s)[\\Delta x(s-\\frac{1}{2}) ]=\\delta_{nm}d_n^2",
"semanticFormula": "\\sum_{s=a}^{b-1} \\dualHahnpolyR{n}@{c}{s}{a}{b} \\dualHahnpolyR{m}@{c}{s}{a}{b} \\rho(s) [\\Delta x(s - \\frac{1}{2})] = \\delta_{nm} d_n^2",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 8
}
],
"includes": [
"n"
],
"isPartOf": [],
"definiens": [
{
"definition": "Dual Hahn polynomial",
"score": 2
}
]
}
]
} |
| 75 | p_n(x;a,b,c|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_n*_4\Phi_3(q^{-n},abcdq^{n-1},ae^{i{(t+2u)}},ae^{-it};abe^{2iu},ac,ad;q;q)
|
Continuous q-Hahn polynomials | Full data:
{
"id": 75,
"pid": 127,
"eid": "math.127.0",
"title": "Continuous q-Hahn polynomials",
"formulae": [
{
"id": "FORMULA_67e28846328978f4e08bb6b69fe6c549",
"formula": "p_n(x;a,b,c|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_n*_4\\Phi_3(q^{-n},abcdq^{n-1},ae^{i{(t+2u)}},ae^{-it};abe^{2iu},ac,ad;q;q)",
"semanticFormula": "p_n(x ; a , b , c|q) = a^{-n} \\expe^{-\\iunit nu} \\qmultiPochhammersym{ab\\expe^{2\\iunit u} , ac , ad}{q}{n} * \\qgenhyperphi{4}{3}@{q^{-n} , abcdq^{n-1} , a\\expe^{\\iunit{(t+2u)}} , a\\expe^{-\\iunit t}}{ab\\expe^{2\\iunit u} , ac , ad}{q}{q}",
"confidence": 0.8662724998444776,
"translations": {
"Mathematica": {
"translation": "p[n_, x_, a_, b_, c_, q_] := (a)^(- n)* Exp[- I*\\[Nu]]*Product[QPochhammer[Part[{a*b*Exp[2*I*u], a*c , a*d},i],q,n],{i,1,Length[{a*b*Exp[2*I*u], a*c , a*d}]}]* QHypergeometricPFQ[{(q)^(- n), a*b*c*d*(q)^(n - 1), a*Exp[I*(t + 2*u)], a*Exp[- I*t]},{a*b*Exp[2*I*u], a*c , a*d},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 1
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "continuous FORMULA_7694f4a66316e53c8cdd9d9954bd611d - Hahn polynomial",
"score": 2
},
{
"definition": "q - Pochhammer symbol",
"score": 2
}
]
}
]
} |
| 76 | p_n(x;a,b,c\mid q)=\frac{(ab,ac;q)_n}{a^n}\cdot {_3\Phi_2}(q^-n,ae^{i\theta},ae^{-i\theta}; ab, ac \mid q;q)
|
Continuous dual q-Hahn polynomials | Full data:
{
"id": 76,
"pid": 128,
"eid": "math.128.0",
"title": "Continuous dual q-Hahn polynomials",
"formulae": [
{
"id": "FORMULA_95daf919f18506606090e49a38d1c1a6",
"formula": "p_n(x;a,b,c\\mid q)=\\frac{(ab,ac;q)_n}{a^n}\\cdot {_3\\Phi_2}(q^-n,ae^{i\\theta},ae^{-i\\theta}; ab, ac \\mid q;q)",
"semanticFormula": "p_n(x ; a , b , c \\mid q) = \\frac{\\qmultiPochhammersym{ab , ac}{q}{n}}{a^n} \\cdot \\qgenhyperphi{3}{2}@{q^- n , ae^{\\iunit \\theta} , ae^{- \\iunit \\theta}}{ab , ac}{q}{q}",
"confidence": 0.8662724998444776,
"translations": {
"Mathematica": {
"translation": "p[n_, x_, a_, b_, c_, q_] := Divide[Product[QPochhammer[Part[{a*b , a*c},i],q,n],{i,1,Length[{a*b , a*c}]}],(a)^(n)] * QHypergeometricPFQ[{(q)^(-)* n , a*(e)^(I*\\[Theta]), a*(e)^(- I*\\[Theta])},{a*b , a*c},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "continuous dual FORMULA_7694f4a66316e53c8cdd9d9954bd611d - Hahn polynomial",
"score": 2
}
]
}
]
} |
| 77 | Q_n(x;a,b,N;q)=\;_{3}\phi_2\left[\begin{matrix} q^-n & abq^n+1 & x \\ aq & q^-N \end{matrix} ; q,q \right]
|
Q-Hahn polynomials | Full data:
{
"id": 77,
"pid": 129,
"eid": "math.129.0",
"title": "Q-Hahn polynomials",
"formulae": [
{
"id": "FORMULA_b3a9ac90714e1e705d2a88b30e79cca0",
"formula": "Q_n(x;a,b,N;q)=\\;_{3}\\phi_2\\left[\\begin{matrix} q^-n & abq^n+1 & x \\\\ aq & q^-N \\end{matrix} ; q,q \\right]",
"semanticFormula": "\\qHahnpolyQ{n}@{x}{a}{b}{N}{q} = \\qgenhyperphi{3}{2}@{q^-n , abq^n+1 , x}{aq , q^-N}{q}{q}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Q[n_, x_, a_, b_, N_, q_] := QHypergeometricPFQ[{(q)^(-)* n , a*b*(q)^(n)+ 1 , x},{a*q , (q)^(-)* N},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "q - Hahn polynomial",
"score": 2
},
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 0
}
]
}
]
} |
| 78 | x=
|
Al-Salam–Chihara polynomials | Full data:
{
"id": 78,
"pid": 131,
"eid": "math.131.0",
"title": "Al-Salam\u2013Chihara polynomials",
"formulae": [
{
"id": "FORMULA_52a07ce46212cbc2298415c5fca6e075",
"formula": "x=",
"semanticFormula": "x=\\cos@{\\theta}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "x = Cos[\\[Theta]]"
},
"Maple": {
"translation": "x = cos(theta)"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 20
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "cosine function",
"score": 2
},
{
"definition": "substitution",
"score": 2
}
]
}
]
} |
| 79 | \Phi_n^*(z)=z^n\overline{\Phi_n(1/\overline{z})}
|
Orthogonal polynomials on the unit circle | Full data:
{
"id": 79,
"pid": 132,
"eid": "math.132.7",
"title": "Orthogonal polynomials on the unit circle",
"formulae": [
{
"id": "FORMULA_f2d41903301a99a3fade5f2f49450694",
"formula": "\\Phi_n^*(z)=z^n\\overline{\\Phi_n(1\/\\overline{z})}",
"semanticFormula": "\\Phi_n^*(z) = z^n{\\conj{\\Phi_n(1 \/ \\conj{z})}}",
"confidence": 0.7579553437219001,
"translations": {
"Mathematica": {
"translation": "\\[CapitalPhi]\\[Prima][n_, z_] := z^n*Conjugate[\\[CapitalPhi][n, Divide[1, Conjugate[z]]]]"
}
},
"positions": [
{
"section": 2,
"sentence": 0,
"word": 8
}
],
"includes": [
"\\Phi_n(z)",
"z^n",
"\\alpha_n"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 2
}
]
}
]
} |
| 80 | P_n(x) = c_n \, \det \begin{bmatrix}m_0 & m_1 & m_2 &\cdots & m_n \\m_1 & m_2 & m_3 &\cdots & m_{n+1} \\&&\vdots&& \vdots \\m_{n-1} &m_n& m_{n+1} &\cdots &m_{2n-1}\\1 & x & x^2 & \cdots & x^n\end{bmatrix}
|
Orthogonal polynomials | Full data:
{
"id": 80,
"pid": 133,
"eid": "math.133.8",
"title": "Orthogonal polynomials",
"formulae": [
{
"id": "FORMULA_c0641714ec593f58211623652c4a34f0",
"formula": "P_n(x) = c_n \\, \\det \\begin{bmatrix}m_0 & m_1 & m_2 &\\cdots & m_n \\\\m_1 & m_2 & m_3 &\\cdots & m_{n+1} \\\\&&\\vdots&& \\vdots \\\\m_{n-1} &m_n& m_{n+1} &\\cdots &m_{2n-1}\\\\1 & x & x^2 & \\cdots & x^n\\end{bmatrix}",
"semanticFormula": "P_n(x) = c_n \\det \\begin{bmatrix}m_0 & m_1 & m_2 &\\cdots & m_n \\\\m_1 & m_2 & m_3 &\\cdots & m_{n+1} \\\\&&\\vdots&& \\vdots \\\\m_{n-1} &m_n& m_{n+1} &\\cdots &m_{2n-1}\\\\1 & x & x^2 & \\cdots & x^n\\end{bmatrix}",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 5,
"sentence": 0,
"word": 16
}
],
"includes": [
"P_{n}",
"c_{n}",
"P_{m}"
],
"isPartOf": [],
"definiens": [
{
"definition": "constant",
"score": 0
},
{
"definition": "normalisation",
"score": 0
},
{
"definition": "orthogonal polynomial",
"score": 2
},
{
"definition": "term of the moment",
"score": 0
}
]
}
]
} |
| 81 | \displaystyle p_n(x;a,b;q) = {}_2\phi_1(q^{-n},abq^{n+1};aq;q,xq)
|
Little q-Jacobi polynomials | Full data:
{
"id": 81,
"pid": 134,
"eid": "math.134.0",
"title": "Little q-Jacobi polynomials",
"formulae": [
{
"id": "FORMULA_c492265e4cd4beeeb776dad843dc1f73",
"formula": "\\displaystyle p_n(x;a,b;q) = {}_2\\phi_1(q^{-n},abq^{n+1};aq;q,xq)",
"semanticFormula": "\\littleqJacobipolyp{n}@{x}{a}{b}{q} = \\qgenhyperphi{2}{1}@{q^{-n} , abq^{n+1}}{aq}{q}{xq}",
"confidence": 0.7229065246531701,
"translations": {
"Mathematica": {
"translation": "p[n_, x_, a_, b_, q_] := QHypergeometricPFQ[{(q)^(- n), a*b*(q)^(n + 1)},{a*q},q,x*q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 19
}
],
"includes": [
"q",
"p_{n}(x;a,b;q)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Jacobi polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 0
},
{
"definition": "little q - Jacobi polynomial",
"score": 2
}
]
}
]
} |
| 82 | \displaystyle P_n(x;a,b,c;q)={}_3\phi_2(q^{-n},abq^{n+1},x;aq,cq;q,q)
|
Big q-Jacobi polynomials | Full data:
{
"id": 82,
"pid": 135,
"eid": "math.135.0",
"title": "Big q-Jacobi polynomials",
"formulae": [
{
"id": "FORMULA_0680f701a101288f89487a7a3fabefb1",
"formula": "\\displaystyle P_n(x;a,b,c;q)={}_3\\phi_2(q^{-n},abq^{n+1},x;aq,cq;q,q)",
"semanticFormula": "\\bigqJacobipolyP{n}@{x}{a}{b}{c}{q} = \\qgenhyperphi{3}{2}@{q^{-n} , abq^{n+1} , x}{aq , cq}{q}{q}",
"confidence": 0.7424814142326033,
"translations": {
"Mathematica": {
"translation": "p[n_, x_, a_, b_, c_, q_] := QHypergeometricPFQ[{(q)^(- n), a*b*(q)^(n + 1), x},{a*q , c*q},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
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}
],
"includes": [
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"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "big q - Jacobi polynomial",
"score": 2
}
]
}
]
} |
| 83 | P_n(x;a,b;q)=\frac{1}{(b^{-1}*q^{-n};q,n)}*_2\Phi_1(q^{-n},aqx^{-1};aq|q;\frac{x}{b})
|
Big q-Laguerre polynomials | Full data:
{
"id": 83,
"pid": 137,
"eid": "math.137.0",
"title": "Big q-Laguerre polynomials",
"formulae": [
{
"id": "FORMULA_aa5a6972c7e8327e316eddc8fd8e9b08",
"formula": "P_n(x;a,b;q)=\\frac{1}{(b^{-1}*q^{-n};q,n)}*_2\\Phi_1(q^{-n},aqx^{-1};aq|q;\\frac{x}{b})",
"semanticFormula": "P_n(x;a,b;q) =\\frac{1}{\\qmultiPochhammersym{b^{-1}*q^{-n}}{q}{n}} * \\qgenhyperphi{2}{1}@{q^{-n},aqx^{-1}}{aq}{q}{\\frac{x}{b}}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "P[n_, x_, a_, b_, q_] := Divide[1,Product[QPochhammer[Part[{(b)^(- 1)* (q)^(- n)},i],q,n],{i,1,Length[{(b)^(- 1)* (q)^(- n)}]}]]* QHypergeometricPFQ[{(q)^(- n), a*q*(x)^(- 1)},{a*q},q,Divide[x,b]]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 1
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "q - Pochhammer symbol",
"score": 1
},
{
"definition": "big q - Laguerre polynomial",
"score": 2
}
]
}
]
} |
| 84 | K_n(\lambda(x);c,N|q)=_3\Phi_2(q^{-n},q^{-x},cq^{x-N};q^{-N},0|q;q)
|
Dual q-Krawtchouk polynomials | Full data:
{
"id": 84,
"pid": 138,
"eid": "math.138.0",
"title": "Dual q-Krawtchouk polynomials",
"formulae": [
{
"id": "FORMULA_9221dfda453868628eb8bbcd2d414fdf",
"formula": "K_n(\\lambda(x);c,N|q)=_3\\Phi_2(q^{-n},q^{-x},cq^{x-N};q^{-N},0|q;q)",
"semanticFormula": "K_n(\\lambda(x);c,N|q) = \\qgenhyperphi{3}{2}@{q^{-n},q^{-x},cq^{x-N}}{q^{-N},0}{q}{q}",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 0
},
{
"definition": "dual q - Krawtchouk polynomial",
"score": 2
}
]
}
]
} |
| 85 | P_{n}^{(\alpha)}(x|q)=\frac{(q^\alpha+1;q)_{n}}{(q;q)_{n}}
|
Continuous q-Laguerre polynomials | Full data:
{
"id": 85,
"pid": 139,
"eid": "math.139.0",
"title": "Continuous q-Laguerre polynomials",
"formulae": [
{
"id": "FORMULA_8c9e3af3c57272f3a6ddabba68ab4d3e",
"formula": "P_{n}^{(\\alpha)}(x|q)=\\frac{(q^\\alpha+1;q)_{n}}{(q;q)_{n}}",
"semanticFormula": "P_{n}^{(\\alpha)}(x|q) = \\frac{\\qmultiPochhammersym{q^\\alpha+1}{q}{n}}{\\qPochhammer{q}{q}{n}} \\qgenhyperphi{3}{2}@{q^{-n},q^{\\alpha\/2+1\/4}\\expe^{\\iunit\\theta},q^{\\alpha\/2+1\/4}*\\expe^{-\\iunit\\theta}}{q^{\\alpha+1},0}{q}{q}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "P[n_, \\[Alpha]_, x_, q_] := Divide[Product[QPochhammer[Part[{(q)^\\[Alpha]+ 1},i],q,n],{i,1,Length[{(q)^\\[Alpha]+ 1}]}],QPochhammer[q, q, n]]*QHypergeometricPFQ[{(q)^(- n), (q)^(\\[Alpha]\/2 + 1\/4)* Exp[I*\\[Theta]], (q)^(\\[Alpha]\/2 + 1\/4)* Exp[- I*\\[Theta]]},{(q)^(\\[Alpha]+ 1), 0},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 1,
"word": 0
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "continuous q - Laguerre polynomial",
"score": 2
},
{
"definition": "family of basic hypergeometric orthogonal polynomial",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 2
}
]
}
]
} |
| 86 | \displaystyle p_n(x;a|q) = {}_2\phi_1(q^{-n},0;aq;q,qx) = \frac{1}{(a^{-1}q^{-n};q)_n}{}_2\phi_0(q^{-n},x^{-1};;q,x/a)
|
Little q-Laguerre polynomials | Full data:
{
"id": 86,
"pid": 142,
"eid": "math.142.0",
"title": "Little q-Laguerre polynomials",
"formulae": [
{
"id": "FORMULA_4e548bca196e13d5af0eaadf2ea725d1",
"formula": "\\displaystyle p_n(x;a|q) = {}_2\\phi_1(q^{-n},0;aq;q,qx) = \\frac{1}{(a^{-1}q^{-n};q)_n}{}_2\\phi_0(q^{-n},x^{-1};;q,x\/a)",
"semanticFormula": "p_n(x ; a|q) = \\qgenhyperphi{2}{1}@{q^{-n} , 0}{aq}{q}{qx} = \\frac{1}{\\qmultiPochhammersym{a^{-1} q^{-n}}{q}{n}} \\qgenhyperphi{2}{0}@{q^{-n} , x^{-1}}{}{q}{x\/a}",
"confidence": 0.7219509974881755,
"translations": {
"Mathematica": "p[n_, x_, a_, q_] := QHypergeometricPFQ[{(q)^(- n), 0},{a*q},q,q*x] == Divide[1,Product[QPochhammer[Part[{(a)^(- 1)* (q)^(- n)},i],q,n],{i,1,Length[{(a)^(- 1)* (q)^(- n)}]}]]*QHypergeometricPFQ[{(q)^(- n), (x)^(- 1)},{},q,x\/a]"
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"q",
"p_{n}(x;a|q)"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "little q - Laguerre polynomial",
"score": 2
},
{
"definition": "q - Pochhammer symbol",
"score": 2
}
]
}
]
} |
| 87 | y_{n}(x;a;q)=\;_{2}\phi_1 \left(\begin{matrix} q^{-N} & -aq^{n} \\ 0 \end{matrix} ; q,qx \right)
|
Q-Bessel polynomials | Full data:
{
"id": 87,
"pid": 143,
"eid": "math.143.0",
"title": "Q-Bessel polynomials",
"formulae": [
{
"id": "FORMULA_c89da2fda6f9f6411ed4292f6d845f52",
"formula": "y_{n}(x;a;q)=\\;_{2}\\phi_1 \\left(\\begin{matrix} q^{-N} & -aq^{n} \\\\ 0 \\end{matrix} ; q,qx \\right)",
"semanticFormula": "y_{n}(x;a;q) = \\qgenhyperphi{2}{1}@{q^{-N} , -aq^{n}}{0}{q}{qx}",
"confidence": 0.6264217257193126,
"translations": {
"Mathematica": "y[n_, x_, a_, q_] := QHypergeometricPFQ[{(q)^(- N), - a*(q)^(n)},{0},q,q*x]"
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 16
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 0
},
{
"definition": "q - Bessel polynomial",
"score": 1
}
]
}
]
} |
| 88 | h_n(ix;q^{-1}) = i^n\hat h_n(x;q)
|
Discrete q-Hermite polynomials | Full data:
{
"id": 88,
"pid": 144,
"eid": "math.144.2",
"title": "Discrete q-Hermite polynomials",
"formulae": [
{
"id": "FORMULA_b9974285610b7a82c94b6a504726df8c",
"formula": "h_n(ix;q^{-1}) = i^n\\hat h_n(x;q)",
"semanticFormula": "\\discqHermitepolyhI{n}@{\\iunit x}{q^{-1}} = \\iunit^n \\discqHermitepolyhII{n}@{x}{q}",
"confidence": 0.8429359579302446,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 27
}
],
"includes": [
"\\hat{h}_{n}(x;q)",
"q",
"h_{n}(x;q)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Hermite polynomial",
"score": 2
},
{
"definition": "term of basic hypergeometric function",
"score": 1
},
{
"definition": "Carlitz polynomial",
"score": 1
},
{
"definition": "Al-Salam",
"score": 1
},
{
"definition": "discrete q - Hermite polynomial",
"score": 2
}
]
}
]
} |
| 89 | P_{n}(x;a\mid q) = a^{-n} e^{in\phi} \frac{a^2;q_n}{(q;q)_n} {_3}\Phi_2(q^-n, ae^{i(\theta+2\phi)}, ae^{-i\theta}; a^2, 0 \mid q; q)
|
Q-Meixner–Pollaczek polynomials | Full data:
{
"id": 89,
"pid": 145,
"eid": "math.145.0",
"title": "Q-Meixner\u2013Pollaczek polynomials",
"formulae": [
{
"id": "FORMULA_fa6650cad7aed4d975716018ef03068f",
"formula": "P_{n}(x;a\\mid q) = a^{-n} e^{in\\phi} \\frac{a^2;q_n}{(q;q)_n} {_3}\\Phi_2(q^-n, ae^{i(\\theta+2\\phi)}, ae^{-i\\theta}; a^2, 0 \\mid q; q)",
"semanticFormula": "P_{n}(x ; a \\mid q) = a^{-n} \\expe^{\\iunit n\\phi} \\frac{\\qmultiPochhammersym{a^2}{q}{n}}{\\qmultiPochhammersym{q}{q}{n}} \\qgenhyperphi{3}{2}@{q^- n , a\\expe^{\\iunit(\\theta + 2 \\phi)} , a\\expe^{- \\iunit \\theta}}{a^2, 0}{q}{q}",
"confidence": 0.8662724998444776,
"translations": {
"Mathematica": {
"translation": "P[n_, x_, a_, q_] := (a)^(- n)* Exp[I*n*\\[Phi]]*Divide[Product[QPochhammer[Part[{(a)^(2)},i],q,n],{i,1,Length[{(a)^(2)}]}],Product[QPochhammer[Part[{q},i],q,n],{i,1,Length[{q}]}]]*QHypergeometricPFQ[{(q)^(-)* n , a*Exp[I*(\\[Theta]+ 2*\\[Phi])], a*Exp[- I*\\[Theta]]},{(a)^(2), 0},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 16
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "q - Pochhammer symbol",
"score": 2
},
{
"definition": "Q Meixner \u2013 Pollaczek polynomials",
"score": 2
}
]
}
]
} |
| 90 | \displaystyle L_n^{(\alpha)}(x;q) = \frac{(q^{\alpha+1};q)_n}{(q;q)_n} {}_1\phi_1(q^{-n};q^{\alpha+1};q,-q^{n+\alpha+1}x)
|
Q-Laguerre polynomials | Full data:
{
"id": 90,
"pid": 149,
"eid": "math.149.0",
"title": "Q-Laguerre polynomials",
"formulae": [
{
"id": "FORMULA_dea0af895f73964b98741e71bc0635cb",
"formula": "\\displaystyle L_n^{(\\alpha)}(x;q) = \\frac{(q^{\\alpha+1};q)_n}{(q;q)_n} {}_1\\phi_1(q^{-n};q^{\\alpha+1};q,-q^{n+\\alpha+1}x)",
"semanticFormula": "\\qLaguerrepolyL{\\alpha}{n}@{x}{q} = \\frac{\\qmultiPochhammersym{q^{\\alpha+1}}{q}{n}}{\\qmultiPochhammersym{q}{q}{n}} \\qgenhyperphi{1}{1}@{q^{-n}}{q^{\\alpha+1}}{q}{- q^{n+\\alpha+1} x}",
"confidence": 0.779734956061429,
"translations": {
"Mathematica": {
"translation": "L[n_, \\[Alpha]_, x_, q_] := Divide[Product[QPochhammer[Part[{(q)^(\\[Alpha]+ 1)},i],q,n],{i,1,Length[{(q)^(\\[Alpha]+ 1)}]}],Product[QPochhammer[Part[{q},i],q,n],{i,1,Length[{q}]}]]*QHypergeometricPFQ[{(q)^(- n)},{(q)^(\\[Alpha]+ 1)},q,- (q)^(n + \\[Alpha]+ 1)* x]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 18
}
],
"includes": [
"q",
"P_{n}^{(\\alpha)}(x;q)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Laguerre polynomial",
"score": 2
},
{
"definition": "q - Laguerre polynomial",
"score": 2
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "q - Pochhammer symbol",
"score": 2
}
]
}
]
} |
| 91 | \sum_{n=0}^\infty H_n(x \mid q) \frac{t^n}{(q;q)_n} = \frac{1}{\left( t e^{i \theta},t e^{-i \theta};q \right)_\infty}
|
Continuous q-Hermite polynomials | Full data:
{
"id": 91,
"pid": 150,
"eid": "math.150.3",
"title": "Continuous q-Hermite polynomials",
"formulae": [
{
"id": "FORMULA_a10dc9de9b2b618ad2f2e96dc9eb0207",
"formula": "\\sum_{n=0}^\\infty H_n(x \\mid q) \\frac{t^n}{(q;q)_n} = \\frac{1}{\\left( t e^{i \\theta},t e^{-i \\theta};q \\right)_\\infty}",
"semanticFormula": "\\sum_{n=0}^\\infty \\contqHermitepolyH{n}@{x}{q} \\frac{t^n}{\\qmultiPochhammersym{q}{q}{n}} = \\frac{1}{\\qmultiPochhammersym{t \\expe^{\\iunit \\theta} , t \\expe^{- \\iunit \\theta}}{q}{\\infty}}",
"confidence": 0.7796357038819148,
"translations": {},
"positions": [
{
"section": 3,
"sentence": 0,
"word": 0
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "continuous q - Hermite polynomial",
"score": 2
},
{
"definition": "q - Pochhammer symbol",
"score": 2
}
]
}
]
} |
| 92 | w^{\prime\prime}+\xi\sin(2z)w^{\prime}+(\eta-p\xi\cos(2z))w=0.
|
Ince equation | Full data:
{
"id": 92,
"pid": 151,
"eid": "math.151.0",
"title": "Ince equation",
"formulae": [
{
"id": "FORMULA_ce9ed9f979f486263028e3d86b63ac60",
"formula": "w^{\\prime\\prime}+\\xi\\sin(2z)w^{\\prime}+(\\eta-p\\xi\\cos(2z))w=0. ",
"semanticFormula": "w^{\\prime\\prime}+\\xi\\sin(2z)w^{\\prime}+(\\eta-p\\xi\\cos(2z))w=0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "D[w[z], {z, 2}] + \\[Xi]*Sin[2*z]*D[w[z], {z, 1}] + (\\[Eta]-p*\\[Xi]*Cos[2*z])*w[z] == 0"
}
},
"positions": [
{
"section": 0,
"sentence": 0,
"word": 19
}
],
"includes": [
"p"
],
"isPartOf": [],
"definiens": [
{
"definition": "differential equation",
"score": 2
},
{
"definition": "Ince equation",
"score": 2
},
{
"definition": "mathematics",
"score": 0
},
{
"definition": "non-negative integer",
"score": 0
},
{
"definition": "Edward Lindsay Ince",
"score": 0
},
{
"definition": "polynomial solution",
"score": 0
},
{
"definition": "Ince polynomial",
"score": 1
}
]
}
]
} |
| 93 | Q_v^\mu(x)= \cos(\mu\pi)\left(\frac{1+x}{1-x}\right)^{\mu/2}\frac{F(v+1,-v;1-\mu;1/2-2/x)} {\Gamma(1-\mu ) }
|
Ferrers function | Full data:
{
"id": 93,
"pid": 152,
"eid": "math.152.1",
"title": "Ferrers function",
"formulae": [
{
"id": "FORMULA_b5ab87b9cd2da05be00884345889d9e3",
"formula": "Q_v^\\mu(x)= \\cos(\\mu\\pi)\\left(\\frac{1+x}{1-x}\\right)^{\\mu\/2}\\frac{F(v+1,-v;1-\\mu;1\/2-2\/x)} {\\Gamma(1-\\mu ) }",
"semanticFormula": "\\FerrersQ[\\mu]{v}@{x} = \\cos(\\mu \\cpi)(\\frac{1+x}{1-x})^{\\mu\/2} \\frac{\\hyperF@{v+1}{-v}{1-\\mu}{1\/2-2\/x}}{\\EulerGamma@{1-\\mu}}",
"confidence": 0.8133162393162393,
"translations": {
"Mathematica": {
"translation": "LegendreQ[v, \\[Mu], x] == Cos[(\\[Mu]*Pi)*]*(Divide[1 + x,1 - x])^(\\[Mu]\/2)*Divide[Hypergeometric2F1[v + 1, - v, 1 - \\[Mu], 1\/2 - 2\/x],Gamma[1 - \\[Mu]]]"
},
"Maple": {
"translation": "LegendreQ(v, mu, x) = cos((mu*Pi)*)*((1 + x)\/(1 - x))^(mu\/2)*(hypergeom([v + 1, - v], [1 - mu], 1\/2 - 2\/x))\/(GAMMA(1 - mu))"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 13
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "Ferrers function of the second kind",
"score": 2
},
{
"definition": "Ferrers function of the first kind",
"score": 1
},
{
"definition": "Gamma function",
"score": 2
},
{
"definition": "hypergeometric function",
"score": 2
}
]
}
]
} |
| 94 | H_{-v}^{(1)}(z,w)=e^{v\pi i}H_v^{(1)}(z,w)
|
Incomplete Bessel functions | Full data:
{
"id": 94,
"pid": 153,
"eid": "math.153.27",
"title": "Incomplete Bessel functions",
"formulae": [
{
"id": "FORMULA_35ab66efafff0de40d98c0778ebb63c3",
"formula": "H_{-v}^{(1)}(z,w)=e^{v\\pi i}H_v^{(1)}(z,w)",
"semanticFormula": "H_{-v}^{(1)}(z,w) = \\expe^{v \\cpi \\iunit} H_v^{(1)}(z , w)",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 2,
"sentence": 0,
"word": 16
}
],
"includes": [
"v",
"w",
"H_v^{(1)}(z,w)"
],
"isPartOf": [],
"definiens": [
{
"definition": "incomplete Bessel function",
"score": 2
}
]
}
]
} |
| 95 | K_v(x,y)=\int_1^\infty\frac{e^{-xt-\frac{y}{t}}}{t^{v+1}}dt
|
Incomplete Bessel K function/generalized incomplete gamma function | Full data:
{
"id": 95,
"pid": 154,
"eid": "math.154.0",
"title": "Incomplete Bessel K function\/generalized incomplete gamma function",
"formulae": [
{
"id": "FORMULA_c333a7966510ed0b8f4de3147eabe47a",
"formula": "K_v(x,y)=\\int_1^\\infty\\frac{e^{-xt-\\frac{y}{t}}}{t^{v+1}}dt",
"semanticFormula": "K_v(x , y) = \\int_1^\\infty \\frac{\\expe^{-xt-\\frac{y}{t}}}{t^{v+1}} \\diff{t}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "K[v_, x_, y_] := Integrate[Divide[Exp[- x*t -Divide[y,t]],(t)^(v + 1)], {t, 1, Infinity}, GenerateConditions->None]"
}
},
"positions": [
{
"section": 0,
"sentence": 0,
"word": 18
}
],
"includes": [
"K_v(x,y)"
],
"isPartOf": [],
"definiens": [
{
"definition": "mathematician",
"score": 0
},
{
"definition": "type incomplete-version of Bessel function",
"score": 2
},
{
"definition": "type generalized-version of incomplete gamma function",
"score": 0
}
]
}
]
} |