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

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

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

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

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')