Search results
Results From The WOW.Com Content Network
The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles. The gamma function has no zeros, so the reciprocal gamma function 1 / Γ(z) is an entire function.
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 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 ) !
Euler's product formula for the gamma function, combined with the functional equation and an identity for the Euler–Mascheroni constant, yields the following expression for the digamma function, valid in the complex plane outside the negative integers (Abramowitz and Stegun 6.3.16): [1]
The notation γ appears nowhere in the writings of either Euler or Mascheroni, and was chosen at a later time, perhaps because of the constant's connection to the gamma function. [3] For example, the German mathematician Carl Anton Bretschneider used the notation γ in 1835, [ 4 ] and Augustus De Morgan used it in a textbook published in parts ...
Contour plot of the beta function. 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.
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:
In integral calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely e i x {\displaystyle e^{ix}} and e − i x {\displaystyle e^{-ix}} and then integrated.