Euler hypergeometric integral

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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<\Re b< \Re c. 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.

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