# Series and integral transforms in Python

Let $$z$$ be a complex element in the unit disc. Let's consider an integral transform from $$0$$ to $$\infty$$ denoted by $$R_{n,m}(f)(z)$$ with two fixed parameters $$n$$ and $$m$$ where $$f$$ is a square integrable function on $$\mathbb{R^+}$$.

I wish to draw graph in $$3D$$ in general for $$n=0,m=0$$ and $$n=1,m=0$$ and $$n=0,m=1$$ and $$n=1,m=1$$ and $$n=1,m=2$$ and $$n=2,m=1$$ and $$n=2,m=2$$ and $$n=2,m=3$$ and $$n=3,m=2$$ and $$n=3,m=3$$.

For example

• Image for Bessel functions

• Image Generalized Laguerre polynomials

Same problem for infinite series, I have tried the following program just for $$m=1$$ and $$n=1$$ but it showed an error.

import matplotlib.pyplot as plt
import sympy as sym
import math
import numpy as np

from mpl_toolkits.mplot3d import Axes3D

x=np.linespace(0.0,0.5,20)
y=np.linespace(0.0,0.5,20)

n=1
m=1
z=complex(x,y)

def zero_to_infinity():
i = 0

while True:
yield i
i += 1

def CalcMYSeries(x,y):
res, temp = 0, 0
for i in zero_to_infinity():
res += ((math.gamma(i+n+m))*((z)**i) )
if res == temp:
break
temp = res
return res

f=CalcMYSeries(x,y)

X, Y= np.meshgrid(x, y)

F=f(X,Y)

fig=plt.figure(figsize=[12,8])
ax = fig.gca(projection = '3d')
ax.plot_surface(X, Y, F, cmap=cm.coolwarm)

plt.show()


I have tried this many times but it didn't work. How to plot this series $$K_{n,m}(z)$$ for $$z$$ in the complex unit disc? Moreover, how to write modified Bessel function in python?