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Although the main definition of the gamma function—the Euler integral of the second kind—is only valid (on the real axis) for positive arguments, its domain can be extended with analytic continuation [13] to negative arguments by shifting the negative argument to positive values by using either the Euler's reflection formula ...
In mathematics, a reflection formula or reflection relation for a function f is a relationship between f(a − x) and f(x). It is a special case of a functional equation . It is common in mathematical literature to use the term "functional equation" for what are specifically reflection formulae.
The gamma function is an important special function in mathematics.Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general.
The classical gamma function satisfies the functional equation (+) = for any .This has an analogue with respect to the Morita gamma function: (+) = {,,.The Euler's reflection formula () = has its following simple counterpart in the p-adic case:
the gamma function, a generalization of the factorial [2] the upper incomplete gamma function; the modular group, the group of fractional linear transformations; the gamma distribution, a continuous probability distribution defined using the gamma function; second-order sensitivity to price in mathematical finance
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
In mathematics, the Barnes G-function G(z) is a function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician Ernest William Barnes. [1] It can be written in terms of the double gamma function.
The Euler integral of the second kind is the gamma function [2] = For positive integers m and n , the two integrals can be expressed in terms of factorials and binomial coefficients : B ( n , m ) = ( n − 1 ) !