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import math

from scipy.integrate import quad

from numpy import inf

def gamma(num: float) -> float:

"""

https://en.wikipedia.org/wiki/Gamma_function

In mathematics, the gamma function is one commonly

used extension of the factorial function to complex numbers.

The gamma function is defined for all complex numbers except the non-positive

integers

>>> gamma(-1)

Traceback (most recent call last):

...

ValueError: math domain error

>>> gamma(0)

Traceback (most recent call last):

...

ValueError: math domain error

>>> gamma(9)

40320.0

>>> from math import gamma as math_gamma

>>> all(.99999999 < gamma(i) / math_gamma(i) <= 1.000000001

... for i in range(1, 50))

True

>>> from math import gamma as math_gamma

>>> gamma(-1)/math_gamma(-1) <= 1.000000001

Traceback (most recent call last):

...

ValueError: math domain error

>>> from math import gamma as math_gamma

>>> gamma(3.3) - math_gamma(3.3) <= 0.00000001

True

"""

if num <= 0:

raise ValueError("math domain error")

return quad(integrand, 0, inf, args=(num))[0]

def integrand(x: float, z: float) -> float:

return math.pow(x, z - 1) * math.exp(-x)

if __name__ == "__main__":

from doctest import testmod

testmod()

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