Barnes Integral

In mathematics, a Barnes integral or Mellin–Barnes integral is a contour integral involving a product of gamma functions. They were introduced by . They are closely related to generalized hypergeometric series.

The integral is usually taken along a contour which is a deformation of the imaginary axis passing to the right of all poles of factors of the form Γ(''a'' + ''s'') and to the left of all poles of factors of the form Γ(''a'' − ''s'').

Hypergeometric series

The hypergeometric function is given as a Barnes integral by

:$_2F_1(a,b;c;z) =\frac \frac \int_^ \frac(-z)^s\,ds,$

see also . This equality can be obtained by moving the contour to the right while picking up the residues at ''s'' = 0, 1, 2, ... . for $z\ll 1$, and by analytic continuation elsewhere. Given proper convergence conditions, one can relate more general Barnes' integrals and generalized hypergeometric functions _''p''''F''_''q'' in a similar way .

Barnes lemmas

The first Barnes lemma states

:$\frac \int_^ \Gamma(a+s)\Gamma(b+s)\Gamma(c-s)\Gamma(d-s)ds =\frac.$

This is an analogue of Gauss's ₂''F''₁ summation formula, and also an extension of Euler's beta integral. The integral in it is sometimes called Barnes's beta integral.

The second Barnes lemma states

:$\frac \int_^ \fracds$

:$=\frac$

where ''e'' = ''a'' + ''b'' + ''c'' − ''d'' + 1. This is an analogue of Saalschütz's summation formula.

q-Barnes integrals

There are analogues of Barnes integrals for basic hypergeometric series, and many of the other results can also be extended to this case .

References

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* (there is a 2008 paperback with {{ISBN, 978-0-521-09061-2)

Special functions
Hypergeometric functions