Matplotlib and Scipy
====================
I'm preparing a math presentation about fractional calculus, and adding
some graphs to it is a great idea.
Fractional calculus uses the Gamma function to generalize the factorial.
If you want to use the gamma function, the full path the it is

  scipy.special.gamma

To draw a smooth graph, The function scipy.interpolate.make_interp_spline(x,y)
exists:

  X_Y_Spline = scipy.interpolate.make_interp_spline(x,y)

That's a function that returns a function named X_Y_Spline, and with the
returned function, you can define an array of Y values as follows:

  Y_ = X_Y_Spline(x)

In the next line, I take a color from a cmap that is defined as the 'HSV'
map:

  color=cmap(alpha/2)

and then plot using:

  ax.plot(x,Y_,color=color)

Following is the full code of the script:

#!/usr/bin/env python3.11

import matplotlib.pyplot as plt
import matplotlib as mpl
import scipy
import numpy as np

alphas=np.arange(0,1.1,0.1)
x=np.linspace(0,1,1000)
cmap=mpl.colormaps['hsv']
fig,ax = plt.subplots(1,figsize=[5,5])
ax.set_title(r'$J^{\alpha}f(x)$ for some values of $\alpha$ where f(x)=1')

for alpha in alphas:
  y=x**alpha/scipy.special.gamma(alpha+1)
  X_Y_Spline = scipy.interpolate.make_interp_spline(x,y)
  Y_ = X_Y_Spline(x)
  color=cmap(alpha/2)
  ax.plot(x,Y_,color=color)
  ax.text(0.95,0.45-alpha/2,np.round(alpha,1),size=14,color=color)
plt.show()


=== End of code ===
The 'HSV' color map is good to show a gradual change.