Gamma function

extension of the factorial function, with its argument shifted down by 1, to real and complex numbers
The gamma function along part of the real axis

In mathematics, the gamma function (Γ(z)) is an extension of the factorial function to all complex numbers except negative integers. For positive integers, it is defined as

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:


Particular valuesEdit

Some particular values of the gamma function are:


Pi functionEdit

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.


Analytic number theoryEdit

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 in 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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