#!/usr/bin/python
# -*- coding: utf8 -*-

import matplotlib.pyplot as plt
import matplotlib as mpl
import numpy as np
from math import *

code_website = 'http://commons.wikimedia.org/wiki/User:Geek3/mplwp'
try:
    import mplwp
except ImportError, er:
    print 'ImportError:', er
    print 'You need to download mplwp.py from', code_website
    exit(1)

name = 'mplwp_factorial_gamma_stirling.svg'
fig = mplwp.fig_standard(mpl)

xlim = -0.2, 3.2; fig.gca().set_xlim(xlim)
ylim = 0, 6.5; fig.gca().set_ylim(ylim)
mplwp.mark_axeszero(fig.gca())

from scipy.special import gamma
from scipy.optimize import brentq

f1 = lambda x: gamma(1 + x)
x1 = range(4)
y1 = [f1(xx) for xx in x1]
plt.plot(x1, y1, 'o', label='n!', zorder=3)

x2max = brentq(lambda x: f1(x) - ylim[1], 0, xlim[1])
x2 = np.linspace(xlim[0], x2max, 5000)
y2 = [f1(xx) for xx in x2]
plt.plot(x2, y2, label='$\Gamma(n+1)$', zorder=2,
         color=fig.gca().lines[-1].get_color())

f3 = lambda x: sqrt(2*pi*x)*(x/e)**x
x3max = brentq(lambda x: f3(x) - ylim[1], 0, xlim[1])
x3 = np.linspace(0, x3max, 5000)
y3 = [f3(xx) for xx in x3]
plt.plot(x3, y3, label=ur'$\sqrt{2\pi n}(n/e)^n$', zorder=1)

mpl.rc('legend', borderaxespad=1.3)
plt.legend(loc='upper left')
plt.savefig(name)
mplwp.postprocess(name)