gamma Algorithm
In the International Phonetic Alphabet and other modern Latin-alphabet based phonetic notations, it represents the voiced velar fricative. In Modern Greek, this letter represents either a voiced velar fricative or a voiced palatal fricative. The Greek letter gamma Γ was derived from the Phoenician letter for the /g/ phoneme (𐤂 gīml), and as such is cognate with Hebrew gimel ג.𐌂, Roman c and G, Runic kaunan ᚲ,In Archaic Greece, the shape of gamma was closer to a classical lambda (Λ), while lambda retained the Phoenician L-shape (𐌋).
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(gamma(i)/math_gamma(i) <= 1.000000001 and abs(gamma(i)/math_gamma(i)) > .99999999 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()
