Gamma function

extension of the factorial function, with its argument shifted down by 1, to real and complex numbers

In mathematics, the gamma function (Γ(z)) is a key topic in the field of special functions. Γ(z) is an extension of the factorial function to all complex numbers except negative integers. For positive integers, it is defined as [1][2]

The gamma function along part of the real axis

The gamma function is defined for all complex numbers, but it is not defined for negative integers and zero. For a complex number whose real part is a positive integer, the function is defined by:[2][3]

Properties change

Particular values change

Some particular values of the gamma function are:


Pi function change

Gauss introduced the Pi function. This is another way of denoting the gamma function. In terms of the gamma function, the Pi function is


so that


for every non-negative integer n.

Applications change

Analytic number theory change

The gamma function is used to study the Riemann zeta function. A property of the Riemann zeta function is its functional equation:


Bernhard Riemann found a relation between these two functions. This was published in his 1859 paper "Über die Anzahl der Primzahlen unter einer gegebenen Grösse" ("On the Number of Prime Numbers less than a Given Quantity")


Related pages change

Notes change

  1. "List of Probability and Statistics Symbols". Math Vault. 2020-04-26. Retrieved 2020-10-05.
  2. 2.0 2.1 Weisstein, Eric W. "Gamma Function". Retrieved 2020-10-05.
  3. "gamma function | Properties, Examples, & Equation". Encyclopedia Britannica. Retrieved 2020-10-05.

References change