Barnes integral

Summary

A Barnes integral, also called a Mellin–Barnes integral, is a kind of contour integral. Its integrand is a quotient of products of values of the gamma function, times a power $z^{-s}$. The path runs from $-i\infty$ to $+i\infty$ and separates two families of poles of the integrand. Many special functions, such as the hypergeometric function, have such an integral as a representation. Closing the path to the left or to the right and summing residues turns the integral into a series. This gives convergent series and asymptotic expansions.