Gamma function
extension of the factorial function, with its argument shifted down by 1, to real and complex numbers From Wikipedia, the free encyclopedia
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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 a positive integer, the function is defined by:[2][3]
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Properties
Particular values
Some particular values of the gamma function are:
Pi function
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.
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Applications
Analytic number theory
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")
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Related pages
- Gamma distribution
Notes
References
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