integral representation of the hypergeometric function

When c>b>0, one has the representation


Note that the conditions on b and c are necessary for the integralDlmfPlanetmath to be convergent at the endpoints 0 and 1. To see that this integral indeed equals the hypergeometric functionDlmfDlmfDlmfMathworldPlanetmath, it suffices to consider the case |z|<1 since both sides of the equation are analytic functionsMathworldPlanetmath of z. (This follows from the rigidity theorem for analytic functions although some care is required because the functionMathworldPlanetmath is multiply-valued.) With this assumptionPlanetmathPlanetmath, |tz|<1 if t is a real number in the interval [0,1] and hence, (1-tz)-a may be expanded in a power seriesMathworldPlanetmath. Substituting this series in the right hand side of the formulaMathworldPlanetmathPlanetmath above gives


Since the series is uniformly convergent, it is permissible to integrate term-by-term. Interchanging integration and summation and pulling constants outside the integral sign, one obtains


The integrals appearing inside the sum are Euler beta functions. Expressing them in terms of gamma functionsDlmfDlmfMathworldPlanetmath and simplifying, one sees that this integral indeed equals the hypergeometric function.

The hypergeometic function is multiply-valued. To obtain different branches of the hypergeometric function, one can vary the path of integration.

Title integral representation of the hypergeometric function
Canonical name IntegralRepresentationOfTheHypergeometricFunction
Date of creation 2013-03-22 14:35:14
Last modified on 2013-03-22 14:35:14
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Theorem
Classification msc 33C05