Jump to content

File:LogGamma Analytic Function.png

Page contents not supported in other languages.
This is a file from the Wikimedia Commons
From Wikipedia, the free encyclopedia

Original file (900 × 600 pixels, file size: 108 KB, MIME type: image/png)

Summary

Description
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

I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
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.

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')

Captions

Add a one-line explanation of what this file represents

Items portrayed in this file

depicts

26 April 2017

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current13:51, 26 April 2017Thumbnail for version as of 13:51, 26 April 2017900 × 600 (108 KB)StsmithUser created page with UploadWizard

The following page uses this file:

Global file usage

The following other wikis use this file:

Metadata