Gamma function

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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:

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:

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.

Contents

Motivation

The gamma function can be seen as a solution to the following interpolation problem:

A plot of the first few factorials makes clear that such a curve can be drawn, but it would be preferable to have a formula that precisely describes the curve, in which the number of operations does not depend on the size of n. The formula for the factorial, n!, cannot be used directly for fractional values of n since it is only valid when n is an integer. There is, in fact, no such simple solution for factorials; any combination of sums, products, powers, exponential functions, or logarithms with a fixed number of terms will not suffice to express n!. However, it is possible to find a general formula for factorials using tools such as integrals and limits from calculus. A good solution to this is the gamma function.

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