In mathematics, the gamma function (represented by the capital Greek letter Γ) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. That is, if n is a positive integer:

n\equiv 0(\mbox{mod k}).

Although the gamma function is defined for all complex numbers except the non-positive integers, it is defined via an improper integral that converges only for complex numbers with a positive real part:

n\equiv 0(\mbox{mod k}).

This integral function is extended by analytic continuation to all complex numbers except the non-positive integers (where the function has simple poles), yielding the meromorphic function we call the gamma function.

The gamma function is a component in various probability-distribution functions, and as such it is applicable in the fields of probability and statistics, as well as combinatorics.

The gamma function along part of the real axis.

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