E8.5The gamma function
The gamma function is related to the factorial by \(\Gamma(x) = (x-1)!\) and both are plotted in the code below. Note that \(\Gamma(x)\) is not defined for negative integer \(x\), which leads to discontinuities in the plot.
gammafunction
import numpy as np
from scipy.special import gamma
import matplotlib.pyplot as plt
# The Gamma function
ax = np.linspace(-5, 5, 1000)
plt.plot(ax, gamma(ax), ls="-", c="k", label="$\Gamma(x)$")
# (x-1)! for x = 1, 2, ..., 6
ax2 = np.linspace(1, 6, 6)
xm1fac = np.array([1, 1, 2, 6, 24, 120])
plt.plot(
ax2,
xm1fac,
marker="*",
markersize=10,
markeredgecolor="r",
markerfacecolor="r",
ls="",
c="r",
label="$(x-1)!$",
)
plt.ylim(-50, 50)
plt.xlim(-5, 5)
plt.xlabel("$x$")
plt.legend()
plt.show()
