Poject:GoldData: Difference between revisions
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! colspan="3" | Entry Info | ! colspan="3" | Entry Info | ||
! colspan="2" | Translations | ! colspan="2" | Translations | ||
! colspan="5" | Reason For Failure | ! colspan="5" | Reason For Failure | ||
! colspan="2" | | ! colspan="2" | | ||
|- | |- | ||
! # !! Formula !! Title !! Semantic LaTeX !! CAS Translations !! Definition / Substitution !! Pattern Matching !! Derivatives / Primes !! Missing Infos !! Untranslatable Macro !! Explanation !! Evaluation Data | ! # !! Formula !! Title !! Semantic LaTeX !! CAS Translations !! Definition / Substitution !! Pattern Matching !! Derivatives / Primes !! Missing Infos !! Untranslatable Macro !! Explanation !! Evaluation Data | ||
|- | |- | ||
| 1 | | [[Gold 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> | | <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]] | | [[Bessel function#math.51.18| Bessel function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
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| 2 | | [[Gold 2]] | ||
| <syntaxhighlight lang="tex" inline >E(e) \,=\, \int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta</syntaxhighlight> | | <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]] | | [[Ellipse#math.52.404| Ellipse]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
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| 3 | | [[Gold 3]] | ||
| <syntaxhighlight lang="tex" inline >F(x;k) = u</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >F(x;k) = u</syntaxhighlight> | ||
| [[Elliptic integral#math.53.6| Elliptic integral]] | | [[Elliptic integral#math.53.6| Elliptic integral]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
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| 4 | | [[Gold 4]] | ||
| <syntaxhighlight lang="tex" inline >\frac{1}{\Gamma(z)} = \frac{i}{2\pi}\int_C (-t)^{-z}e^{-t}\,dt</syntaxhighlight> | | <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]] | | [[Gamma function#math.54.195| Gamma function]] | ||
| {{ya}} | | {{ya}} | ||
| - | | - | ||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
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| 5 | | [[Gold 5]] | ||
| <syntaxhighlight lang="tex" inline >2^{4} = 2 \times2 \times 2 \times 2 = 16</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >2^{4} = 2 \times2 \times 2 \times 2 = 16</syntaxhighlight> | ||
| [[Logarithm#| Logarithm]] | | [[Logarithm#| Logarithm]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 6 | | [[Gold 6]] | ||
| <syntaxhighlight lang="tex" inline >\psi(x) := \sum_{n=1}^\infty e^{-n^2 \pi x}</syntaxhighlight> | | <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]] | | [[Riemann zeta function#math.56.40| Riemann zeta function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 7 | | [[Gold 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> | | <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]] | | [[Logarithmic integral function#math.57.2| Logarithmic integral function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
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| 8 | | [[Gold 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> | | <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]] | | [[Gaussian quadrature#math.58.61| Gaussian quadrature]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 9 | | [[Gold 9]] | ||
| <syntaxhighlight lang="tex" inline >\begin{align}x & =ue^u, \\[5pt]\frac{dx}{du} & =(u+1)e^u.\end{align}</syntaxhighlight> | | <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]] | | [[Lambert W function#math.59.52| Lambert W function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
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| 10 | | [[Gold 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> | | <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]] | | [[Legendre polynomials#math.60.57| Legendre polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 11 | | [[Gold 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> | | <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]] | | [[Error function#math.61.27| Error function]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 12 | | [[Gold 12]] | ||
| <syntaxhighlight lang="tex" inline >x_k = \cos\left(\frac{\pi(k+1/2)}{n}\right),\quad k=0,\ldots,n-1</syntaxhighlight> | | <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]] | | [[Chebyshev polynomials#math.62.44| Chebyshev polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
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| 13 | | [[Gold 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> | | <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]] | | [[Hermite polynomials#math.63.109| Hermite polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
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| 14 | | [[Gold 14]] | ||
| <syntaxhighlight lang="tex" inline >x = \pm 1</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >x = \pm 1</syntaxhighlight> | ||
| [[Legendre function#math.64.8| Legendre function]] | | [[Legendre function#math.64.8| Legendre function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 15 | | [[Gold 15]] | ||
| <syntaxhighlight lang="tex" inline >E_n=2^nE_n(\tfrac{1}{2})</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >E_n=2^nE_n(\tfrac{1}{2})</syntaxhighlight> | ||
| [[Bernoulli polynomials#math.65.27| Bernoulli polynomials]] | | [[Bernoulli polynomials#math.65.27| Bernoulli polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 1,592: | Line 1,592: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 16 | | [[Gold 16]] | ||
| <syntaxhighlight lang="tex" inline >\operatorname{Si}(ix) = i\operatorname{Shi}(x)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\operatorname{Si}(ix) = i\operatorname{Shi}(x)</syntaxhighlight> | ||
| [[Trigonometric integral#math.66.8| Trigonometric integral]] | | [[Trigonometric integral#math.66.8| Trigonometric integral]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 1,683: | Line 1,683: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
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| 17 | | [[Gold 17]] | ||
| <syntaxhighlight lang="tex" inline >f(z)=\frac{1}{\Beta(x,y)}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >f(z)=\frac{1}{\Beta(x,y)}</syntaxhighlight> | ||
| [[Beta function#math.67.29| Beta function]] | | [[Beta function#math.67.29| Beta function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 1,767: | Line 1,767: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 18 | | [[Gold 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> | | <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]] | | [[Fresnel integral#math.68.51| Fresnel integral]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 1,833: | Line 1,833: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 19 | | [[Gold 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> | | <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]] | | [[Classical orthogonal polynomials#math.69.117| Classical orthogonal polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
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}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 20 | | [[Gold 20]] | ||
| <syntaxhighlight lang="tex" inline >{}_1F_0(1;;z) = \sum_{n \geqslant 0} z^n = (1-z)^{-1}</syntaxhighlight> | | <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]] | | [[Generalized hypergeometric function#math.70.58| Generalized hypergeometric function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 1,983: | Line 1,983: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
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| 21 | | [[Gold 21]] | ||
| <syntaxhighlight lang="tex" inline >\chi(-1) = 1</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\chi(-1) = 1</syntaxhighlight> | ||
| [[Dirichlet L-function#math.71.1| Dirichlet L-function]] | | [[Dirichlet L-function#math.71.1| Dirichlet L-function]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 2,063: | Line 2,063: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 22 | | [[Gold 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> | | <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]] | | [[Airy function#math.72.15| Airy function]] | ||
| {{na}} | | {{na}} | ||
| - | | - | ||
| Line 2,143: | Line 2,143: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 23 | | [[Gold 23]] | ||
| <syntaxhighlight lang="tex" inline >F'(y)=1-2yF(y)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >F'(y)=1-2yF(y)</syntaxhighlight> | ||
| [[Dawson function#math.73.41| Dawson function]] | | [[Dawson function#math.73.41| Dawson function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 2,200: | Line 2,200: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 24 | | [[Gold 24]] | ||
| <syntaxhighlight lang="tex" inline >s\not =1</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >s\not =1</syntaxhighlight> | ||
| [[Hurwitz zeta function#math.74.0| Hurwitz zeta function]] | | [[Hurwitz zeta function#math.74.0| Hurwitz zeta function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 2,252: | Line 2,252: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 25 | | [[Gold 25]] | ||
| <syntaxhighlight lang="tex" inline >q = e^{i\pi\tau}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >q = e^{i\pi\tau}</syntaxhighlight> | ||
| [[Theta function#math.75.6| Theta function]] | | [[Theta function#math.75.6| Theta function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 2,317: | Line 2,317: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 26 | | [[Gold 26]] | ||
| <syntaxhighlight lang="tex" inline >\frac{\mathrm{d}}{\mathrm{d}z} \operatorname{dn}(z) = - k^2 \operatorname{sn}(z) \operatorname{cn}(z)</syntaxhighlight> | | <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]] | | [[Jacobi elliptic functions#math.76.155| Jacobi elliptic functions]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 2,391: | Line 2,391: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 27 | | [[Gold 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> | | <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]] | | [[Incomplete gamma function#math.77.118| Incomplete gamma function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 2,466: | Line 2,466: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 28 | | [[Gold 28]] | ||
| <syntaxhighlight lang="tex" inline >_{1}(z) =</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >_{1}(z) =</syntaxhighlight> | ||
| [[Polylogarithm#math.78.0| Polylogarithm]] | | [[Polylogarithm#math.78.0| Polylogarithm]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 2,542: | Line 2,542: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 29 | | [[Gold 29]] | ||
| <syntaxhighlight lang="tex" inline >\int_{-\infty}^\infty \operatorname{sinc}(t) \, e^{-i 2 \pi f t}\,dt = \operatorname{rect}(f)</syntaxhighlight> | | <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]] | | [[Sinc function#math.79.11| Sinc function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 2,613: | Line 2,613: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 30 | | [[Gold 30]] | ||
| <syntaxhighlight lang="tex" inline >N=1</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >N=1</syntaxhighlight> | ||
| [[Exponential integral#math.80.26| Exponential integral]] | | [[Exponential integral#math.80.26| Exponential integral]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 2,670: | Line 2,670: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 31 | | [[Gold 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> | | <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]] | | [[Laguerre polynomials#math.81.84| Laguerre polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 2,742: | Line 2,742: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 32 | | [[Gold 32]] | ||
| <syntaxhighlight lang="tex" inline >c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!}</syntaxhighlight> | ||
| [[Associated Legendre polynomials#math.82.8| Associated Legendre polynomials]] | | [[Associated Legendre polynomials#math.82.8| Associated Legendre polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 2,797: | Line 2,797: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 33 | | [[Gold 33]] | ||
| <syntaxhighlight lang="tex" inline >\mathrm{Gi}(x) = \frac{1}{\pi} \int_0^\infty \sin\left(\frac{t^3}{3} + xt\right)\, dt</syntaxhighlight> | | <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]] | | [[Scorer's function#math.83.3| Scorer's function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 2,883: | Line 2,883: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 34 | | [[Gold 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> | | <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]] | | [[Voigt profile#math.84.31| Voigt profile]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 2,957: | Line 2,957: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 35 | | [[Gold 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> | | <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]] | | [[Lerch zeta function#math.85.57| Lerch zeta function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 3,030: | Line 3,030: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 36 | | [[Gold 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> | | <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]] | | [[Confluent hypergeometric function#math.86.44| Confluent hypergeometric function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 3,101: | Line 3,101: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 37 | | [[Gold 37]] | ||
| <syntaxhighlight lang="tex" inline >\sigma = \pm 1</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\sigma = \pm 1</syntaxhighlight> | ||
| [[Mathieu function#math.87.54| Mathieu function]] | | [[Mathieu function#math.87.54| Mathieu function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 3,178: | Line 3,178: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 38 | | [[Gold 38]] | ||
| <syntaxhighlight lang="tex" inline >\frac{d^2f}{dz^2} + \left(\tilde{a}z^2+\tilde{b}z+\tilde{c}\right)f=0</syntaxhighlight> | | <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]] | | [[Parabolic cylinder function#math.88.0| Parabolic cylinder function]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 3,240: | Line 3,240: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 39 | | [[Gold 39]] | ||
| <syntaxhighlight lang="tex" inline >c=\infty</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >c=\infty</syntaxhighlight> | ||
| [[Painlevé transcendents#math.89.23| Painlevé transcendents]] | | [[Painlevé transcendents#math.89.23| Painlevé transcendents]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 3,305: | Line 3,305: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 40 | | [[Gold 40]] | ||
| <syntaxhighlight lang="tex" inline >c = a + 1</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >c = a + 1</syntaxhighlight> | ||
| [[Hypergeometric function#math.90.7| Hypergeometric function]] | | [[Hypergeometric function#math.90.7| Hypergeometric function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 3,359: | Line 3,359: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 41 | | [[Gold 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> | | <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]] | | [[Barnes G-function#math.91.47| Barnes G-function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 3,451: | Line 3,451: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 42 | | [[Gold 42]] | ||
| <syntaxhighlight lang="tex" inline >192/24 = 8 = 2 \times 4</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >192/24 = 8 = 2 \times 4</syntaxhighlight> | ||
| [[Heun function#math.92.1| Heun function]] | | [[Heun function#math.92.1| Heun function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 3,691: | Line 3,691: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 43 | | [[Gold 43]] | ||
| <syntaxhighlight lang="tex" inline >=2</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >=2</syntaxhighlight> | ||
| [[Gegenbauer polynomials#math.93.0| Gegenbauer polynomials]] | | [[Gegenbauer polynomials#math.93.0| Gegenbauer polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 3,744: | Line 3,744: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 44 | | [[Gold 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> | | <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]] | | [[Basic hypergeometric series#math.94.4| Basic hypergeometric series]] | ||
| {{na}} | | {{na}} | ||
| - | | - | ||
| Line 3,802: | Line 3,802: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 45 | | [[Gold 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> | | <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]] | | [[Whittaker function#math.95.0| Whittaker function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 3,861: | Line 3,861: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 46 | | [[Gold 46]] | ||
| <syntaxhighlight lang="tex" inline >e_1=\tfrac12,\qquad e_2=0,\qquad e_3=-\tfrac12</syntaxhighlight> | | <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]] | | [[Lemniscatic elliptic function#math.96.1| Lemniscatic elliptic function]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 3,916: | Line 3,916: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 47 | | [[Gold 47]] | ||
| <syntaxhighlight lang="tex" inline >\gamma> 0,n-p=m-q> 0</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\gamma> 0,n-p=m-q> 0</syntaxhighlight> | ||
| [[Meijer G-function#math.98.53| Meijer G-function]] | | [[Meijer G-function#math.98.53| Meijer G-function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 3,974: | Line 3,974: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 48 | | [[Gold 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> | | <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]] | | [[3-j symbol#math.99.30| 3-j symbol]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 4,033: | Line 4,033: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 49 | | [[Gold 49]] | ||
| <syntaxhighlight lang="tex" inline >\begin{Bmatrix} i & j & \ell\\ k & m & n \end{Bmatrix}= (\Phi_{i,j}^{k,m})_{\ell,n}</syntaxhighlight> | | <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]] | | [[6-j symbol#math.100.14| 6-j symbol]] | ||
| {{na}} | | {{na}} | ||
| - | | - | ||
| Line 4,099: | Line 4,099: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 50 | | [[Gold 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> | | <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]] | | [[9-j symbol#math.101.32| 9-j symbol]] | ||
| {{na}} | | {{na}} | ||
| - | | - | ||
| Line 4,158: | Line 4,158: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 51 | | [[Gold 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> | | <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]] | | [[Kravchuk polynomials#math.102.5| Kravchuk polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 4,213: | Line 4,213: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 52 | | [[Gold 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> | | <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]] | | [[Kelvin functions#math.103.8| Kelvin functions]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 4,280: | Line 4,280: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 53 | | [[Gold 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> | | <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]] | | [[Lommel function#math.104.2| Lommel function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 4,344: | Line 4,344: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 54 | | [[Gold 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> | | <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]] | | [[Struve function#math.105.18| Struve function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 4,417: | Line 4,417: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 55 | | [[Gold 55]] | ||
| <syntaxhighlight lang="tex" inline >f(t+p) = f(t)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >f(t+p) = f(t)</syntaxhighlight> | ||
| [[Hill differential equation#math.106.7| Hill differential equation]] | | [[Hill differential equation#math.106.7| Hill differential equation]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 4,479: | Line 4,479: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 56 | | [[Gold 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> | | <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]] | | [[Anger function#math.108.3| Anger function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 4,542: | Line 4,542: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 57 | | [[Gold 57]] | ||
| <syntaxhighlight lang="tex" inline >(\operatorname{Ec})^'_{2K} = (\operatorname{Ec})^'_0 = 0, \;\; (\operatorname{Es})^'_{2K} = (\operatorname{Es})^'_0 = 0</syntaxhighlight> | | <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]] | | [[Lamé function#math.109.27| Lamé function]] | ||
| {{ya}} | | {{ya}} | ||
| - | | - | ||
| Line 4,593: | Line 4,593: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 58 | | [[Gold 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> | | <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]] | | [[Gauss–Hermite quadrature#math.110.1| Gauss–Hermite quadrature]] | ||
| {{ya}} | | {{ya}} | ||
| - | | - | ||
| Line 4,658: | Line 4,658: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 59 | | [[Gold 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> | | <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]] | | [[Askey–Wilson polynomials#math.111.0| Askey–Wilson polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 4,718: | Line 4,718: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 60 | | [[Gold 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> | | <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]] | | [[Hahn polynomials#math.112.0| Hahn polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 4,777: | Line 4,777: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 61 | | [[Gold 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> | | <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]] | | [[Charlier polynomials#math.113.2| Charlier polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 4,829: | Line 4,829: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 62 | | [[Gold 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> | | <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]] | | [[Q-Racah polynomials#math.114.0| Q-Racah polynomials]] | ||
| {{na}} | | {{na}} | ||
| - | | - | ||
| Line 4,873: | Line 4,873: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 63 | | [[Gold 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> | | <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]] | | [[Q-Charlier polynomials#math.115.0| Q-Charlier polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| - | | - | ||
| Line 4,921: | Line 4,921: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 64 | | [[Gold 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> | | <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]] | | [[Meixner polynomials#math.116.0| Meixner polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 4,977: | Line 4,977: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 65 | | [[Gold 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> | | <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]] | | [[Appell series#math.117.19| Appell series]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 5,047: | Line 5,047: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 66 | | [[Gold 66]] | ||
| <syntaxhighlight lang="tex" inline >\Theta_\Lambda(\tau) = \sum_{x\in\Lambda}e^{i\pi\tau\|x\|^2}\qquad\mathrm{Im}\,\tau > 0</syntaxhighlight> | | <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]] | | [[Theta function of a lattice#math.118.0| Theta function of a lattice]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 5,105: | Line 5,105: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 67 | | [[Gold 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> | | <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]] | | [[Heine–Stieltjes polynomials#math.119.0| Heine–Stieltjes polynomials]] | ||
| {{na}} | | {{na}} | ||
| - | | - | ||
| Line 5,185: | Line 5,185: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 68 | | [[Gold 68]] | ||
| <syntaxhighlight lang="tex" inline >w(x) = \frac{k}{\sqrt{\pi}} x^{-1/2} \exp(-k^2\log^2 x)</syntaxhighlight> | | <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]] | | [[Stieltjes–Wigert polynomials#math.120.0| Stieltjes–Wigert polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 5,270: | Line 5,270: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 69 | | [[Gold 69]] | ||
| <syntaxhighlight lang="tex" inline >y^2=x(x-1)(x-\lambda)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >y^2=x(x-1)(x-\lambda)</syntaxhighlight> | ||
| [[Modular lambda function#math.121.23| Modular lambda function]] | | [[Modular lambda function#math.121.23| Modular lambda function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 5,331: | Line 5,331: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 70 | | [[Gold 70]] | ||
| <syntaxhighlight lang="tex" inline >P_1^{(\lambda)}(x;\phi)=2(\lambda\cos\phi + x\sin\phi)</syntaxhighlight> | | <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]] | | [[Meixner–Pollaczek polynomials#math.122.3| Meixner–Pollaczek polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| - | | - | ||
| Line 5,377: | Line 5,377: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 71 | | [[Gold 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> | | <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]] | | [[Jacobi polynomials#math.123.0| Jacobi polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 5,444: | Line 5,444: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 72 | | [[Gold 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> | | <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]] | | [[Continuous dual Hahn polynomials#math.124.0| Continuous dual Hahn polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| - | | - | ||
| Line 5,502: | Line 5,502: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 73 | | [[Gold 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> | | <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]] | | [[Continuous Hahn polynomials#math.125.15| Continuous Hahn polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 5,562: | Line 5,562: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 74 | | [[Gold 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> | | <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]] | | [[Dual Hahn polynomials#math.126.7| Dual Hahn polynomials]] | ||
| {{na}} | | {{na}} | ||
| - | | - | ||
| Line 5,608: | Line 5,608: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 75 | | [[Gold 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> | | <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]] | | [[Continuous q-Hahn polynomials#math.127.0| Continuous q-Hahn polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 5,674: | Line 5,674: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 76 | | [[Gold 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> | | <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]] | | [[Continuous dual q-Hahn polynomials#math.128.0| Continuous dual q-Hahn polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 5,736: | Line 5,736: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 77 | | [[Gold 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> | | <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]] | | [[Q-Hahn polynomials#math.129.0| Q-Hahn polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 5,796: | Line 5,796: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 78 | | [[Gold 78]] | ||
| <syntaxhighlight lang="tex" inline >x=</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >x=</syntaxhighlight> | ||
| [[Al-Salam–Chihara polynomials#math.131.0| Al-Salam–Chihara polynomials]] | | [[Al-Salam–Chihara polynomials#math.131.0| Al-Salam–Chihara polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 5,851: | Line 5,851: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 79 | | [[Gold 79]] | ||
| <syntaxhighlight lang="tex" inline >\Phi_n^*(z)=z^n\overline{\Phi_n(1/\overline{z})}</syntaxhighlight> | | <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]] | | [[Orthogonal polynomials on the unit circle#math.132.7| Orthogonal polynomials on the unit circle]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 5,903: | Line 5,903: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 80 | | [[Gold 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> | | <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]] | | [[Orthogonal polynomials#math.133.8| Orthogonal polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| - | | - | ||
| Line 5,963: | Line 5,963: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 81 | | [[Gold 81]] | ||
| <syntaxhighlight lang="tex" inline >\displaystyle p_n(x;a,b;q) = {}_2\phi_1(q^{-n},abq^{n+1};aq;q,xq)</syntaxhighlight> | | <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]] | | [[Little q-Jacobi polynomials#math.134.0| Little q-Jacobi polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 6,026: | Line 6,026: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 82 | | [[Gold 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> | | <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]] | | [[Big q-Jacobi polynomials#math.135.0| Big q-Jacobi polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 6,085: | Line 6,085: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 83 | | [[Gold 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> | | <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]] | | [[Big q-Laguerre polynomials#math.137.0| Big q-Laguerre polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 6,151: | Line 6,151: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 84 | | [[Gold 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> | | <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]] | | [[Dual q-Krawtchouk polynomials#math.138.0| Dual q-Krawtchouk polynomials]] | ||
| {{na}} | | {{na}} | ||
| - | | - | ||
| Line 6,209: | Line 6,209: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 85 | | [[Gold 85]] | ||
| <syntaxhighlight lang="tex" inline >P_{n}^{(\alpha)}(x|q)=\frac{(q^\alpha+1;q)_{n}}{(q;q)_{n}}</syntaxhighlight> | | <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]] | | [[Continuous q-Laguerre polynomials#math.139.0| Continuous q-Laguerre polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 6,267: | Line 6,267: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 86 | | [[Gold 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> | | <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]] | | [[Little q-Laguerre polynomials#math.142.0| Little q-Laguerre polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 6,332: | Line 6,332: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 87 | | [[Gold 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> | | <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]] | | [[Q-Bessel polynomials#math.143.0| Q-Bessel polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 6,390: | Line 6,390: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 88 | | [[Gold 88]] | ||
| <syntaxhighlight lang="tex" inline >h_n(ix;q^{-1}) = i^n\hat h_n(x;q)</syntaxhighlight> | | <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]] | | [[Discrete q-Hermite polynomials#math.144.2| Discrete q-Hermite polynomials]] | ||
| {{na}} | | {{na}} | ||
| - | | - | ||
| Line 6,454: | Line 6,454: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 89 | | [[Gold 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> | | <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]] | | [[Q-Meixner–Pollaczek polynomials#math.145.0| Q-Meixner–Pollaczek polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 6,518: | Line 6,518: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 90 | | [[Gold 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> | | <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]] | | [[Q-Laguerre polynomials#math.149.0| Q-Laguerre polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 6,585: | Line 6,585: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 91 | | [[Gold 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> | | <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]] | | [[Continuous q-Hermite polynomials#math.150.3| Continuous q-Hermite polynomials]] | ||
| {{na}} | | {{na}} | ||
| - | | - | ||
| Line 6,635: | Line 6,635: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 92 | | [[Gold 92]] | ||
| <syntaxhighlight lang="tex" inline >w^{\prime\prime}+\xi\sin(2z)w^{\prime}+(\eta-p\xi\cos(2z))w=0. </syntaxhighlight> | | <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]] | | [[Ince equation#math.151.0| Ince equation]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 6,709: | Line 6,709: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 93 | | [[Gold 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> | | <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]] | | [[Ferrers function#math.152.1| Ferrers function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 6,772: | Line 6,772: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 94 | | [[Gold 94]] | ||
| <syntaxhighlight lang="tex" inline >H_{-v}^{(1)}(z,w)=e^{v\pi i}H_v^{(1)}(z,w)</syntaxhighlight> | | <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]] | | [[Incomplete Bessel functions#math.153.27| Incomplete Bessel functions]] | ||
| {{ya}} | | {{ya}} | ||
| - | | - | ||
| Line 6,820: | Line 6,820: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 95 | | [[Gold 95]] | ||
| <syntaxhighlight lang="tex" inline >K_v(x,y)=\int_1^\infty\frac{e^{-xt-\frac{y}{t}}}{t^{v+1}}dt</syntaxhighlight> | | <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]] | | [[Incomplete Bessel K function/generalized incomplete gamma function#math.154.0| Incomplete Bessel K function/generalized incomplete gamma function]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
Revision as of 14:53, 1 September 2021
Download gold-data.json.
| Entry Info | Translations | Reason For Failure | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Formula | Title | Semantic LaTeX | CAS Translations | Definition / Substitution | Pattern Matching | Derivatives / Primes | Missing Infos | Untranslatable Macro | Explanation | Evaluation Data |
| Gold 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 | 
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- | - | - | - | - | - | Full data:
|
| Gold 2 | E(e) \,=\, \int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta
|
Ellipse | 
|
|
- | 
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- | - | - | e was interpreted as Euler's number
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Full data:
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| Gold 3 | F(x;k) = u
|
Elliptic integral |
|
|
|
- | - | - | - | x is substituted
|
Full data:
|
| Gold 4 | \frac{1}{\Gamma(z)} = \frac{i}{2\pi}\int_C (-t)^{-z}e^{-t}\,dt
|
Gamma function |
|
- | - |
|
- | - | - | Contour integrals cannot be translated. | Full data:
|
| Gold 5 | 2^{4} = 2 \times2 \times 2 \times 2 = 16
|
Logarithm |
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- | - | - | - | - | - | Full data:
|
| Gold 6 | \psi(x) := \sum_{n=1}^\infty e^{-n^2 \pi x}
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Riemann zeta function |
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- | - | - | - | - | - | Full data:
|
| Gold 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 |
|
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- | - | - | - | - | - | Full data:
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| Gold 8 | w_{i} = \frac{1}{p'_{n}(x_{i})}\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx
|
Gaussian quadrature |
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|
- | - |
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- | - | - | Full data:
|
| Gold 9 | \begin{align}x & =ue^u, \\[5pt]\frac{dx}{du} & =(u+1)e^u.\end{align}
|
Lambert W function |
|
|
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- | - | - | - | - | Full data:
|
| Gold 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:
|
| Gold 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 |
|
|
- | - |
|
- | - | was not detected as k-th derivative but as power. | Full data:
|
| Gold 12 | x_k = \cos\left(\frac{\pi(k+1/2)}{n}\right),\quad k=0,\ldots,n-1
|
Chebyshev polynomials |
|
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- | - | - | - | - | - | Full data:
|
| Gold 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 |
|
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- | - | - | - | - | - | Full data:
|
| Gold 14 | x = \pm 1
|
Legendre function |
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- | - | - | - | - | - | Full data:
|
| Gold 15 | E_n=2^nE_n(\tfrac{1}{2})
|
Bernoulli polynomials |
|
|
- |
|
- | - | - | Both E where detected as Euler's number but the second is Euler polynomial. | Full data:
|
| Gold 16 | \operatorname{Si}(ix) = i\operatorname{Shi}(x)
|
Trigonometric integral |
|
|
- | - | - | - | - | Integral was not tagged as a noun by CoreNLP. Hence, the macro for hyperbolic sine function was retrieved too late and not considered for replacements. | Full data:
|
| Gold 17 | f(z)=\frac{1}{\Beta(x,y)}
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Beta function |
|
|
|
- | - |
|
- | The original formula contained f(z) but should have been f(x,z). This was fixed in the Wikipedia article after we generated the dataset.
|
Full data:
|
| Gold 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 |
|
|
- |
|
- | - | - | Matrix argument of does not exist in the DLMF. | Full data:
|
| Gold 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 |
|
|
- | - | - |
|
- | No info about Gamma function. | Full data:
|
| Gold 20 | {}_1F_0(1;;z) = \sum_{n \geqslant 0} z^n = (1-z)^{-1}
|
Generalized hypergeometric function |
|
|
- |
|
- | - | - | Empty arguments did not match the semantic macros (bug). | Full data:
|
| Gold 21 | \chi(-1) = 1
|
Dirichlet L-function |
|
|
- | - | - |
|
- | It was translated to DirichletCharacter[1, k, - 1] == 1. The only valid input for k is 1.
|
Full data:
|
| Gold 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 |
|
- | - | - |
|
- | - | No translation possible for \sim
|
Full data:
|
| Gold 23 | F'(y)=1-2yF(y)
|
Dawson function |
|
|
- | - |
|
|
- | No dependency to Dawson. | Full data:
|
| Gold 24 | s\not =1
|
Hurwitz zeta function |
|
|
- | - | - | - | - | - | Full data:
|
| Gold 25 | q = e^{i\pi\tau}
|
Theta function |
|
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- | - | - | - | - | - | Full data:
|
| Gold 26 | \frac{\mathrm{d}}{\mathrm{d}z} \operatorname{dn}(z) = - k^2 \operatorname{sn}(z) \operatorname{cn}(z)
|
Jacobi elliptic functions |
|
|
- | - | - | - | - | - | Full data:
|
| Gold 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:
|
| Gold 28 | _{1}(z) =
|
Polylogarithm |
|
|
- | - | - | - | - | Wrong math detection. | Full data:
|
| Gold 29 | \int_{-\infty}^\infty \operatorname{sinc}(t) \, e^{-i 2 \pi f t}\,dt = \operatorname{rect}(f)
|
Sinc function |
|
|
- | - | - | - | - | - | Full data:
|
| Gold 30 | N=1
|
Exponential integral |
|
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- | - | - | - | - | - | Full data:
|
| Gold 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 |
|
|
- | - | - |
|
- | No infos about the gamma function. | Full data:
|
| Gold 32 | c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!}
|
Associated Legendre polynomials |
|
|
- | - | - | - | - | - | Full data:
|
| Gold 33 | \mathrm{Gi}(x) = \frac{1}{\pi} \int_0^\infty \sin\left(\frac{t^3}{3} + xt\right)\, dt
|
Scorer's function |
|
|
- | - | - | - | - | - | Full data:
|
| Gold 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:
|
| Gold 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 |
|
|
- | - | - | - |
|
Landau notation. | Full data:
|
| Gold 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:
|
| Gold 37 | \sigma = \pm 1
|
Mathieu function |
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- | - | - | - | - | - | Full data:
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| Gold 38 | \frac{d^2f}{dz^2} + \left(\tilde{a}z^2+\tilde{b}z+\tilde{c}\right)f=0
|
Parabolic cylinder function |
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|
- | - | - |
|
- | ODE. f does not show the argument z. | Full data:
|
| Gold 39 | c=\infty
|
Painlevé transcendents |
|
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- | - | - | - | - | - | Full data:
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| Gold 40 | c = a + 1
|
Hypergeometric function |
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- | - | - | - | - | - | Full data:
|
| Gold 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 |
|
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- | - | - | - | - | - | Full data:
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| Gold 42 | 192/24 = 8 = 2 \times 4
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Heun function |
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- | - | - | - | - | - | Full data:
|
| Gold 43 | =2
|
Gegenbauer polynomials |
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- | - | - | - | - | Wrong math detection. | Full data:
|
| Gold 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 |
|
- | - |
|
- | - | - | Indef length of arguments are not translatable. | Full data:
|
| Gold 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:
|
| Gold 46 | e_1=\tfrac12,\qquad e_2=0,\qquad e_3=-\tfrac12
|
Lemniscatic elliptic function |
|
|
- | - | - | - | - | Multi-equation problem (bug). | Full data:
|
| Gold 47 | \gamma> 0,n-p=m-q> 0
|
Meijer G-function |
|
|
- | - | - | - | - | - | Full data:
|
| Gold 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 |
|
|
- | - | - | - | - | LCT does not support matrix translations yet. | Full data:
|
| Gold 49 | \begin{Bmatrix} i & j & \ell\\ k & m & n \end{Bmatrix}= (\Phi_{i,j}^{k,m})_{\ell,n}
|
6-j symbol |
|
- | - | - | - |
|
- | A matrix cannot be defined as a function in Mathematica. | Full data:
|
| Gold 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 |
|
- | - | - | - | - | - | Mistakenly interpreted as Wigner 6-j rather than 9-j. | Full data:
|
| Gold 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 |
|
|
- | - | - |
|
- | Krawtchouk vs Kravchuk (synonym problem) | Full data:
|
| Gold 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:
|
| Gold 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 |
|
|
- | - | - |
|
- | No information about gamma function | Full data:
|
| Gold 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 |
|
|
- |
|
- | - | - | Arguments of are split by commas. That is wrong notation. Hence, our semantic patterns did not match. | Full data:
|
| Gold 55 | f(t+p) = f(t)
|
Hill differential equation |
|
|
- | - | - | - | - | - | Full data:
|
| Gold 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 |
|
|
- | - | - |
|
- | No information about gamma function. | Full data:
|
| Gold 57 | (\operatorname{Ec})^'_{2K} = (\operatorname{Ec})^'_0 = 0, \;\; (\operatorname{Es})^'_{2K} = (\operatorname{Es})^'_0 = 0
|
Lamé function |
|
- | - | - | - | - | - | No translation possible. | Full data:
|
| Gold 58 | \int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)
|
Gauss–Hermite quadrature |
|
- | - | - | - | - | - | No translation possible. | Full data:
|
| Gold 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 |
|
|
|
- | - | - | - | Could not extract the name Askey-Wilson polynomials. | Full data:
|
| Gold 60 | Q_n(x;\alpha,\beta,N)= {}_3F_2(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1).
|
Hahn polynomials |
|
|
|
- | - | - |
|
- | Full data:
|
| Gold 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 |
|
|
- | - | - |
|
- | Did not found Charlier polynomial. | Full data:
|
| Gold 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 |
|
- | - | - | - | - |
|
Did not find q-Recah polynomial. Since it is not a definition, and q-Recah are not supported by Mathematica, there is no translation possible. | Full data:
|
| Gold 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:
|
| Gold 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 |
|
|
- | - | - |
|
- | Did not find Meixner. | Full data:
|
| Gold 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 |
|
|
- |
|
- |
|
- | Cannot match hidden arguments of Appell F1 function. | Full data:
|
| Gold 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:
|
| Gold 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 |
|
- | - |
|
- | - | - | Mistakenly detected Stieltjes constant. No translation possible for S. | Full data:
|
| Gold 68 | w(x) = \frac{k}{\sqrt{\pi}} x^{-1/2} \exp(-k^2\log^2 x)
|
Stieltjes–Wigert polynomials |
|
|
|
- | - | - | - | - | Full data:
|
| Gold 69 | y^2=x(x-1)(x-\lambda)
|
Modular lambda function |
|
|
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| Gold 70 | P_1^{(\lambda)}(x;\phi)=2(\lambda\cos\phi + x\sin\phi)
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Meixner–Pollaczek polynomials |
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| Gold 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)
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Jacobi polynomials |
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| Gold 72 | S_n(x^2;a,b,c)= {}_3F_2(-n,a+ix,a-ix;a+b,a+c;1).
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Continuous dual Hahn polynomials |
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| Gold 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)
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Continuous Hahn polynomials |
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| Gold 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
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Dual Hahn polynomials |
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| Gold 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)
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Continuous q-Hahn polynomials |
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| Gold 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)
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Continuous dual q-Hahn polynomials |
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Underscore mismatch. | Full data:
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| Gold 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]
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Q-Hahn polynomials |
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| Gold 78 | x=
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Al-Salam–Chihara polynomials |
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| Gold 79 | \Phi_n^*(z)=z^n\overline{\Phi_n(1/\overline{z})}
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Orthogonal polynomials on the unit circle |
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| Gold 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}
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Orthogonal polynomials |
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| Gold 81 | \displaystyle p_n(x;a,b;q) = {}_2\phi_1(q^{-n},abq^{n+1};aq;q,xq)
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Little q-Jacobi polynomials |
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No translation for \littleJacobipolyp
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Full data:
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| Gold 82 | \displaystyle P_n(x;a,b,c;q)={}_3\phi_2(q^{-n},abq^{n+1},x;aq,cq;q,q)
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Big q-Jacobi polynomials |
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No translation for \bigqJacobipolyP
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Full data:
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| Gold 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})
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Big q-Laguerre polynomials |
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| Gold 84 | K_n(\lambda(x);c,N|q)=_3\Phi_2(q^{-n},q^{-x},cq^{x-N};q^{-N},0|q;q)
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Dual q-Krawtchouk polynomials |
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| Gold 85 | P_{n}^{(\alpha)}(x|q)=\frac{(q^\alpha+1;q)_{n}}{(q;q)_{n}}
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Continuous q-Laguerre polynomials |
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| Gold 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)
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Little q-Laguerre polynomials |
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| Gold 87 | y_{n}(x;a;q)=\;_{2}\phi_1 \left(\begin{matrix} q^{-N} & -aq^{n} \\ 0 \end{matrix} ; q,qx \right)
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Q-Bessel polynomials |
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| Gold 88 | h_n(ix;q^{-1}) = i^n\hat h_n(x;q)
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Discrete q-Hermite polynomials |
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Full data:
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| Gold 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)
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Q-Meixner–Pollaczek polynomials |
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Full data:
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| Gold 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)
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Q-Laguerre polynomials |
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| Gold 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}
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Continuous q-Hermite polynomials |
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| Gold 92 | w^{\prime\prime}+\xi\sin(2z)w^{\prime}+(\eta-p\xi\cos(2z))w=0.
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Ince equation |
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| Gold 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 ) }
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Ferrers function |
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| Gold 94 | H_{-v}^{(1)}(z,w)=e^{v\pi i}H_v^{(1)}(z,w)
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Incomplete Bessel functions |
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Full data:
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| Gold 95 | K_v(x,y)=\int_1^\infty\frac{e^{-xt-\frac{y}{t}}}{t^{v+1}}dt
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Incomplete Bessel K function/generalized incomplete gamma function |
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