gamma function.

The gamma function is defined as

for \(\Re(z) > 0\) and is extended to the rest of the complex plane by analytic continuation. See [dlmf] for more details.

Real or complex valued argument

Optional output array for the function values

Values of the gamma function

Notes

The gamma function is often referred to as the generalized factorial since \(\Gamma(n + 1) = n!\) for natural numbers \(n\) . More generally it satisfies the recurrence relation \(\Gamma(z + 1) = z \cdot \Gamma(z)\) for complex \(z\) , which, combined with the fact that \(\Gamma(1) = 1\) , implies the above identity for \(z = n\) .

References

NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/5.2#E1

Examples

>>> import numpy as np
>>> from scipy.special import gamma, factorial
>>> gamma([0, 0.5, 1, 5])
array([         inf,   1.77245385,   1.        ,  24.        ])
>>> z = 2.5 + 1j
>>> gamma(z)
(0.77476210455108352+0.70763120437959293j)
>>> gamma(z+1), z*gamma(z)  # Recurrence property
((1.2292740569981171+2.5438401155000685j),
 (1.2292740569981158+2.5438401155000658j))
>>> gamma(0.5)**2  # gamma(0.5) = sqrt(pi)
3.1415926535897927
Plot gamma(x) for real x
>>> x = np.linspace(-3.5, 5.5, 2251)
>>> y = gamma(x)
>>> import matplotlib.pyplot as plt
>>> plt.plot(x, y, 'b', alpha=0.6, label='gamma(x)')
>>> k = np.arange(1, 7)
>>> plt.plot(k, factorial(k-1), 'k*', alpha=0.6,
...          label='(x-1)!, x = 1, 2, ...')
>>> plt.xlim(-3.5, 5.5)
>>> plt.ylim(-10, 25)
>>> plt.grid()
>>> plt.xlabel('x')
>>> plt.legend(loc='lower right')
>>> plt.show()