Gamma function
Extension of the factorial function / From Wikipedia, the free encyclopedia
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In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n,
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Fields of application | Calculus, mathematical analysis, statistics, physics |
Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral:
The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.[clarification needed]
The gamma function has no zeros, so the reciprocal gamma function 1/Γ(z) is an entire function. In fact, the gamma function corresponds to the Mellin transform of the negative exponential function:
Other extensions of the factorial function do exist, but the gamma function is the most popular and useful. It 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 interpolation problem of finding a smooth curve that connects the points of the factorial sequence: for all positive integer values of . The simple formula for the factorial, x! = 1 × 2 × ⋯ × x is only valid when x is a positive integer, and no elementary function has this property, but a good solution is the gamma function .[1]
The gamma function is not only smooth but analytic (except at the non-positive integers), and it can be defined in several explicit ways. However, it is not the only analytic function that extends the factorial, as one may add any analytic function that is zero on the positive integers, such as for an integer .[1] Such a function is known as a pseudogamma function, the most famous being the Hadamard function.[2]
A more restrictive requirement is the functional equation which interpolates the shifted factorial :[3][4]
But this still does not give a unique solution, since it allows for multiplication by any periodic function with and , such as . One way to resolve the ambiguity is the Bohr–Mollerup theorem, which shows that is the unique interpolating function over the positive reals which is logarithmically convex (super-convex),[5] meaning that is convex.[6]