Euler hypergeometric integral

Euler hypergeometric integral

In mathematics, the Euler hypergeometric integral is a representation of the hypergeometric function by means of an integral. It is given by

:;_2F_1(a,b;c;z)=frac{Gamma(c)}{Gamma(b) Gamma(c-b)}int_0^1 frac{dw} {w^{1-b} ; (1-w)^{1-c+b} ;(1-zw)^a}

which is valid for 0. Note that the conditions on b and c are necessary for the integral to be convergent at the endpoints 0 and 1.

The hypergeometric function is multivalued. Other representations, corresponding to other branches, are given by taking the same integrand, but taking the path of integration to be a closed Pochhammer cycle enclosing the singularities in various orders. Such paths correspond to the monodromy action, and are described in the article hypergeometric differential equation.

ee also

* Schwarz-Christoffel mapping

References

* cite book| author=Milton Abramowitz and Irene A. Stegun, eds.
title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
publisher=Dover, New York |year= 1972 |id=
"(See [http://www.math.sfu.ca/~cbm/aands/page_556.htm Chapter 15] and in particular [http://www.math.sfu.ca/~cbm/aands/page_558.htm Section 15.3] )"
* "(Contains a short proof of the equivalence of the integral representation and the series representation.)"


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