Consolidate gamma (#9769)

* refactor(gamma): Append _iterative to func name

* refactor(gamma): Consolidate implementations

* refactor(gamma): Redundant test function removal

* Update maths/gamma.py

---------

Co-authored-by: Tianyi Zheng <[email protected]>

maths/gamma.py

@@ -1,35 +1,43 @@
+"""
+Gamma function is a very useful tool in math and physics.
+It helps calculating complex integral in a convenient way.
+for more info: 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
+Python's Standard Library math.gamma() function overflows around gamma(171.624).
+"""
 import math
 
 from numpy import inf
 from scipy.integrate import quad
 
 
-def gamma(num: float) -> float:
+def gamma_iterative(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)
+    Calculates the value of Gamma function of num
+    where num is either an integer (1, 2, 3..) or a half-integer (0.5, 1.5, 2.5 ...).
+
+    >>> gamma_iterative(-1)
     Traceback (most recent call last):
         ...
     ValueError: math domain error
-    >>> gamma(0)
+    >>> gamma_iterative(0)
     Traceback (most recent call last):
         ...
     ValueError: math domain error
-    >>> gamma(9)
+    >>> gamma_iterative(9)
     40320.0
     >>> from math import gamma as math_gamma
-    >>> all(.99999999 < gamma(i) / math_gamma(i) <= 1.000000001
+    >>> all(.99999999 < gamma_iterative(i) / math_gamma(i) <= 1.000000001
     ...     for i in range(1, 50))
     True
-    >>> gamma(-1)/math_gamma(-1) <= 1.000000001
+    >>> gamma_iterative(-1)/math_gamma(-1) <= 1.000000001
     Traceback (most recent call last):
         ...
     ValueError: math domain error
-    >>> gamma(3.3) - math_gamma(3.3) <= 0.00000001
+    >>> gamma_iterative(3.3) - math_gamma(3.3) <= 0.00000001
     True
     """
     if num <= 0:
@@ -42,7 +50,66 @@ def integrand(x: float, z: float) -> float:
     return math.pow(x, z - 1) * math.exp(-x)
 
 
+def gamma_recursive(num: float) -> float:
+    """
+    Calculates the value of Gamma function of num
+    where num is either an integer (1, 2, 3..) or a half-integer (0.5, 1.5, 2.5 ...).
+    Implemented using recursion
+    Examples:
+    >>> from math import isclose, gamma as math_gamma
+    >>> gamma_recursive(0.5)
+    1.7724538509055159
+    >>> gamma_recursive(1)
+    1.0
+    >>> gamma_recursive(2)
+    1.0
+    >>> gamma_recursive(3.5)
+    3.3233509704478426
+    >>> gamma_recursive(171.5)
+    9.483367566824795e+307
+    >>> all(isclose(gamma_recursive(num), math_gamma(num))
+    ...     for num in (0.5, 2, 3.5, 171.5))
+    True
+    >>> gamma_recursive(0)
+    Traceback (most recent call last):
+        ...
+    ValueError: math domain error
+    >>> gamma_recursive(-1.1)
+    Traceback (most recent call last):
+        ...
+    ValueError: math domain error
+    >>> gamma_recursive(-4)
+    Traceback (most recent call last):
+        ...
+    ValueError: math domain error
+    >>> gamma_recursive(172)
+    Traceback (most recent call last):
+        ...
+    OverflowError: math range error
+    >>> gamma_recursive(1.1)
+    Traceback (most recent call last):
+        ...
+    NotImplementedError: num must be an integer or a half-integer
+    """
+    if num <= 0:
+        raise ValueError("math domain error")
+    if num > 171.5:
+        raise OverflowError("math range error")
+    elif num - int(num) not in (0, 0.5):
+        raise NotImplementedError("num must be an integer or a half-integer")
+    elif num == 0.5:
+        return math.sqrt(math.pi)
+    else:
+        return 1.0 if num == 1 else (num - 1) * gamma_recursive(num - 1)
+
+
 if __name__ == "__main__":
     from doctest import testmod
 
     testmod()
+    num = 1.0
+    while num:
+        num = float(input("Gamma of: "))
+        print(f"gamma_iterative({num}) = {gamma_iterative(num)}")
+        print(f"gamma_recursive({num}) = {gamma_recursive(num)}")
+        print("\nEnter 0 to exit...")