- or, how eˣ can be approximated by fractions having small degrees of numerator/denominator

Testing Padé approximants

#!/usr/bin/env python
#-*- coding: utf-8 -*-

## Import common moduli
from __future__ import division
import matplotlib, sys, os, time
import matplotlib.pyplot as plt
import numpy as np
from scipy.constants import c, hbar, pi

## Use LaTeX
matplotlib.rc('text', usetex=True)
matplotlib.rc('font', size=8)
matplotlib.rc('text.latex', preamble = '\usepackage{amsmath}, \usepackage{yfonts}, \usepackage{txfonts}, \usepackage{lmodern},')

x=np.linspace(-1,3, 100)
plt.plot(x,np.exp(x), c='k')

# m = 0
plt.plot(x,(1)/(np.ones_like(x))                , c='r', alpha=1)
plt.plot(x,(1)/(1 - x)                          , c='r', alpha=.7)
plt.plot(x,(1)/(1 - x + (1/2)*x**2)             , c='r', alpha=.4)
plt.plot(x,(1)/(1 - x + (1/2)*x**2 - (1/6)*x**3), c='r', alpha=.2)

# m = 1
plt.plot(x,(1 + x)/(1)                                           , c='g', alpha=1)
plt.plot(x,(1 + (1/2)*x)/(1 - (1/2)*x)                           , c='g', alpha=.7)
plt.plot(x,(1 + (1/3)*x)/(1 - (2)/(3)*x + (1)/(6)*x**2)          , c='g', alpha=.5)
plt.plot(x,(1 + (1/4)*x)/(1 - (3/4)*x + (1)/(4)*x**2 - (1/24)*x**3), c='g', alpha=.4)

# m = 3
plt.plot(x,(1 + x + (1/2)*x**2 + (1)/(6)*x**3)/(1)                                            , c='b', alpha=1)
plt.plot(x,(1 + (3/4)*x + (1)/(4)*x**2 + (1/24)*x**3)/(1 - (1/4)*x)                             , c='b', alpha=.7)
plt.plot(x,(1 + (3/5)*x + (3)/(20)*x**2 + (1)/(60)*x**3)/(1 - (2/5)*x + (1/20)*x**2)            , c='b', alpha=.5)
plt.plot(x,(1 + (1/2)*x + (1)/(10)*x**2 + (1)/(120)*x**3)/(1 - (3/7)*x + (1)/(10)*x**2 - (1)/(120)*x**3), c='b', alpha=.4)

# m = 4
plt.plot(x,(1 + x + (1/2)*x**2 + (1)/(6)*x**3+ (1/24)*x**4)/(1)                                                  , c='c', alpha=1)
plt.plot(x,(1 + (4/5)*x + (3/10)*x**2 + (1)/(15)*x**3+ (1)/(120)*x**4)/(1 - (1/5)*x)                               , c='c', alpha=.7)
plt.plot(x,(1 + (2/3)*x + (1/5)*x**2 + (1)/(30)*x**3+ (1)/(360)*x**4)/(1 - (1/3)*x + (1)/(30)*x**2)                , c='c', alpha=.5)
plt.plot(x,(1 + (4/7)*x + (1/7)*x**2 + (2)/(105)*x**3+ (1)/(840)*x**4)/(1 - (3/7)*x + (1)/(14)*x**2 - (1/210)*x**3), c='c', alpha=.4)

## Simple axes
plt.ylim((-1,5)); plt.yscale('linear')
#plt.xlim((-0.1,1.1)); plt.xscale('linear')

## ==== Outputting ====
## Finish the plot + save 
plt.xlabel(u"x"); 
plt.ylabel(u"y"); 
plt.grid()
plt.legend(prop={'size':10}, loc='upper right')
plt.savefig("output.png", bbox_inches='tight')