Gamma function

In mathematics, the gamma function (represented by the capital Greek alphabet 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:

The gamma function is defined for all complex numbers except the non-positive integers. For complex numbers with a positive real part, it is defined via a convergent improper integral:

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. In fact the gamma function corresponds to the Mellin transform of the negative exponential 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 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 $x$ . The simple formula for the factorial, $x! = 1 \times 2 \times \dots \times x$ , cannot be used directly for fractional values of $x$ since it is only valid when $x$ is a natural number (i.e., a positive integer). There are, relatively speaking, no such simple solutions for factorials; no finite combination of sums, products, powers, exponential functions, or logarithms will suffice to express $x!$ ; but 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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