Physics new code (#4709)

* added gamma_function

* Add files via upload

* Resolved issue with str.format

And also changed output to math notation

* Update gamma_function.py

* Rename physics/gamma_function.py to maths/gamma_recursive.py

* Fixes: #4709 Fixed issues for pre-commit test

* Fixes: #4709 solved issues with doctests

And comments

* Fixes: #4709 Added failed tests to doctest

* Align with Python's Standard Library math.gamma()

Replicate the exceptions of https://docs.python.org/3/library/math.html#math.gamma

* Update gamma_recursive.py

* Update gamma_recursive.py

* Update gamma_recursive.py

* Update gamma_recursive.py

Co-authored-by: Christian Clauss <cclauss@me.com>

maths/gamma_recursive.py

@@ -0,0 +1,78 @@
+"""
+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
+
+Python's Standard Library math.gamma() function overflows around gamma(171.624).
+"""
+from math import pi, sqrt
+
+
+def gamma(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(0.5)
+    1.7724538509055159
+    >>> gamma(2)
+    1.0
+    >>> gamma(3.5)
+    3.3233509704478426
+    >>> gamma(171.5)
+    9.483367566824795e+307
+    >>> all(isclose(gamma(num), math_gamma(num)) for num in (0.5, 2, 3.5, 171.5))
+    True
+    >>> gamma(0)
+    Traceback (most recent call last):
+        ...
+    ValueError: math domain error
+    >>> gamma(-1.1)
+    Traceback (most recent call last):
+        ...
+    ValueError: math domain error
+    >>> gamma(-4)
+    Traceback (most recent call last):
+        ...
+    ValueError: math domain error
+    >>> gamma(172)
+    Traceback (most recent call last):
+        ...
+    OverflowError: math range error
+    >>> gamma(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 sqrt(pi)
+    else:
+        return 1.0 if num == 1 else (num - 1) * gamma(num - 1)
+
+
+def test_gamma() -> None:
+    """
+    >>> test_gamma()
+    """
+    assert gamma(0.5) == sqrt(pi)
+    assert gamma(1) == 1.0
+    assert gamma(2) == 1.0
+
+
+if __name__ == "__main__":
+    from doctest import testmod
+
+    testmod()
+    num = 1
+    while num:
+        num = float(input("Gamma of: "))
+        print(f"gamma({num}) = {gamma(num)}")
+        print("\nEnter 0 to exit...")