File:LogGamma Analytic Function.png
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Summary
DescriptionLogGamma Analytic Function.png |
English: Logarithmic Gamma Function Modulus and Phase; the analytic log-Gamma function |
Date | |
Source |
Own work This plot was created with Matplotlib. |
Author | stsmith |
Licensing
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Python Code
import matplotlib as mpl import matplotlib.pyplot as plt from matplotlib import cm from mpl_toolkits.mplot3d import Axes3D import numpy as np import scipy.special as sps # Import lighting object for shading surface plots. from matplotlib.colors import LightSource # Legible plot style defaults # http://matplotlib.org/api/matplotlib_configuration_api.html # http://matplotlib.org/users/customizing.html mpl.rcParams['figure.figsize'] = (10.0, 5.0) mpl.rc('font',**{'family': 'sans-serif', 'weight': 'bold', 'size': 14}) mpl.rc('axes',**{'titlesize': 20, 'titleweight': 'bold', 'labelsize': 16, 'labelweight': 'bold'}) mpl.rc('legend',**{'fontsize': 14}) mpl.rc('figure',**{'titlesize': 16, 'titleweight': 'bold'}) mpl.rc('lines',**{'linewidth': 2.5, 'markersize': 18, 'markeredgewidth': 0}) mpl.rc('mathtext',**{'fontset': 'custom', 'rm': 'sans:bold', 'bf': 'sans:bold', 'it': 'sans:italic', 'sf': 'sans:bold', 'default': 'it'}) # plt.rc('text',usetex=False) # [default] usetex should be False mpl.rcParams['text.latex.preamble'] = [r'\usepackage{amsmath,sfmath} \boldmath'] realmin = np.finfo(np.double).tiny # Define grid of points. xpoints = np.linspace(-3.5, 6, int(np.round((6+3.5)*20))+1) ypoints = np.linspace(-4, 4, (4+4)*20+1) X, Y = np.meshgrid(xpoints, ypoints) # n.b. np.gammaln, np.log(sps.gamma(X+Y*1j)) branch cuts are messed up # e.g. xf = scipy.optimize.minimize(lambda x: np.abs(np.log(sps.gamma(x[0]+x[1]*1j))), np.array([4.,4.]), method='Nelder-Mead', tol=1.e-12) F = sps.loggamma(X+Y*1j) M = np.abs(F) # n.b. this is phase of -log(gamma(z)), not log(gamma(z)) P = np.arctan2(-F.imag,-F.real) nanscale = 1.33 M = np.where(np.isnan(M),nanscale*np.nanmax(M),M) P = np.where(np.isnan(P),0.,P) # Create an hsv array H = (P+np.pi)/(2*np.pi) S = np.ones_like(H) V = 1.-(M-M.min())/(M.max()-M.min()) # Set view parameters for all subplots. azimuth = 290 altitude = 41 # Create empty figure. fig = plt.figure(figsize=(9,6)) # n.b. 1-hsv colors the colors the phase of -log(gamma(z)) correctly w.r.t. hsv cmap facecolors = 1.-mpl.colors.hsv_to_rgb(np.dstack((H,S,V))) f = 0.25 facecolors = f + (1-f)*facecolors # light = LightSource(azimuth+20, altitude-10) light = LightSource(120, 20) illuminated_surface = light.shade_rgb(facecolors, M) # Create a subplot with 3d plotting capabilities. # This command will fail if Axes3D was not imported. ax = fig.add_subplot(111, projection='3d') ax.view_init(altitude, azimuth) ax.plot_surface(X, Y, M, rstride=1, cstride=1, linewidth=0, antialiased=False, facecolors=illuminated_surface, shade=True) plt.xlabel('X',labelpad=10) plt.ylabel('iY',labelpad=10) plt.title('$\log\,\Gamma(z)$') ax.set_zlabel('Modulus',labelpad=5) ax.set_xlim([xpoints.min(), xpoints.max()]) ax.set_ylim([ypoints.min(), ypoints.max()]) ax.set_zlim([0, np.floor(M.max()/nanscale/2)*2]) ax.grid(False) ax.text(0.5, 0.25, 1.5, r'$0! = 1$', (1,0.5,0.5)) ax.text(1.5, 0.5, 1.25, r'$1! = 1$', (1,0.5,0.5)) cax = fig.add_axes([0.9, 0.25, 0.015, 0.5]) cb = mpl.colorbar.ColorbarBase(cax, cmap=plt.cm.hsv, spacing='proportional', ticks=[0, 0.5, 1]) cb.ax.set_yticklabels(['$-\pi$', '$0$', '$\pi$']) cb.set_label('Phase',labelpad=-10) plt.savefig('./loggamma.png')
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26 April 2017
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 14:51, 26 April 2017 | 900 × 600 (108 KB) | wikimediacommons>Stsmith | User created page with UploadWizard |
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