Klein's j-function

In this IPython Notebook we illustrate the Klein's j-function, evaluated in Python mpmath, and plotted with matplotlib, using different color functions for domain coloring method: the default color function for cplot in mpmath (based on HSL color model) or user-defined color functions, such as color_hsv, color_phase (phaseplot), and color_cont (log modulus contour plot).

We list below the color functions to be set in mpmath.cplot:



In [1]:

    
import matplotlib.pyplot as plt 
from mpmath import * 
from colorsys import hsv_to_rgb, hls_to_rgb



In [2]:

    
plt.rcParams['figure.figsize'] = 6, 6



In [3]:

    
pi = 3.1415926535898



In [4]:

    
def hue(z): # hue value corresponding to a complex number
    h = (float(arg(z)) + pi) / (2*pi) 
    return (h + 0.5) % 1.0



In [5]:

    
def color_hsv (fz):
    if isinf(fz):
        return (1.0, 1.0, 1.0)
    if isnan(fz):
        return (0.5, 0.5, 0.5)
    h=hue(fz)
    m=fabs(fz)
    v=(1-1.0/(1+m**2))**0.2#brightness value    
    return  hsv_to_rgb(h, 0.9, v)# s=0.9 is saturation



In [6]:

    
def color_phase (fz):
    if isinf(fz):
        return (1, 1, 1)
    elif isnan(fz):
        return (0.5, 0.5, 0.5)
    else: 
        return  hsv_to_rgb(hue(fz), 1, 1)



In [7]:

    
def PerFract(x, t, m, M):
    x=x/t
    return m+(M-m)*(x-floor(x))  

def color_cont(fz):  
    if isinf(fz):
        return (1, 1, 1)
    if isnan(fz):
        return (0.5, 0.5, 0.5)
    
    n=12  #n is the number of rays drawn in a cycle
    
    h=hue(fz)
    modul=fabs(fz)
    Logm=log(modul) 
    v=PerFract(Logm, 2*pi/n, 0.7, 1)

    return  hsv_to_rgb(h, 1, v)

Another color function we used to plot the Riemann zeta function is defined in this IPython Notebook.

Now calling the function cplot:

cplot(lambda q: kleinj(qbar=q), [-1,1], [-1,1],color=color_function, verbose=True,\ points=100000, file='FileName.png')

cplot(lambda t: kleinj(tau=t), [-1,2], [0,1.5], color=color_function,verbose=True,\ points=100000, file='ImageName.png')

with color_function replaced by one of the above defined color functions one gets an image of Klein's j-function, like those displayed below.

For each plot the keyword points in cplot is set to 100000, except for kleinjCont.png where points=200000.

The images displayed below can be downloaded from github.



In [8]:

    
from IPython.display import Image



In [9]:

    
Image(filename='Imags/kleinjHLS.png')









    Out[9]:



In [10]:

    
Image(filename='Imags/kleinjHSV.png')









    Out[10]:



In [11]:

    
Image(filename='Imags/kleinjPhase.png')









    Out[11]:



In [12]:

    
Image(filename='Imags/kleinjCont.png')









    Out[12]:



In [13]:

    
Image(filename='Imags/kleinjtauHLS.png')









    Out[13]:



In [14]:

    
Image(filename='Imags/kleinjtauHSV.png')









    Out[14]:



In [15]:

    
Image(filename='Imags/kleinjtauPhase.png')









    Out[15]:



In [16]:

    
Image(filename='Imags/kleinjtauCont.png')









    Out[16]:

We posted on Wikipedia similar plots for the following special functions:



In [1]:

    
from IPython.core.display import HTML
def  css_styling():
    styles = open("./custom.css", "r").read()
    return HTML(styles)
css_styling()









    Out[1]: