Gamma function

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The gamma function along part of the real axis

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

Properties[change | change source]

Particular values[change | change source]

Some particular values of the gamma function are:

Pi function[change | change source]

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 | change source]

Analytic number theory[change | change source]

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 | change source]

Notes[change | change source]

  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". mathworld.wolfram.com. Retrieved 2020-10-05.
  3. "gamma function | Properties, Examples, & Equation". Encyclopedia Britannica. Retrieved 2020-10-05.

References[change | change source]