Summary

DescriptionLogGamma Analytic Function.png	English: Logarithmic Gamma Function Modulus and Phase; the analytic log-Gamma function ${\displaystyle \log \Gamma (z)}$
Date	26 April 2017
Source	Own work   This plot was created with Matplotlib.
Author	stsmith

Licensing

I, the copyright holder of this work, hereby publish it under the following license:

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• to remix – to adapt the work

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https://creativecommons.org/licenses/by-sa/4.0CC BY-SA 4.0 Creative Commons Attribution-Share Alike 4.0 truetrue

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')