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In mathematics, a Barnes integral or Mellin–Barnes integral is a contour integral involving a product of gamma functions. They were introduced by Ernest William Barnes (1908, 1910). 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 left of all poles of factors of the form Γ(a + s) and to the right of all poles of factors of the form Γ(a − s).

Hypergeometric series

The hypergeometric function is given as a Barnes integral (Barnes 1908) by

$${}_2F_1(a,b;c;z) =\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \frac{\Gamma(a+s)\Gamma(b+s)\Gamma(-s)}{\Gamma(c+s)}(-z)^s\,ds.$$

This equality can be obtained by moving the contour to the right while picking up the residues at s = 0, 1, 2, ... . Given proper convergence conditions, one can relate more general Barnes' integrals and generalized hypergeometric functions pFq in a similar way.

Barnes lemmas

The first Barnes lemma (Barnes 1908) states

$$\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \Gamma(a+s)\Gamma(b+s)\Gamma(c-s)\Gamma(d-s)ds =\frac{\Gamma(a+c)\Gamma(a+d)\Gamma(b+c)\Gamma(b+d)}{\Gamma(a+b+c+d)}.$$

This is an analogue of Gauss's 2F1 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 (Barnes 1910) states

$$\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \frac{\Gamma(a+s)\Gamma(b+s)\Gamma(c+s)\Gamma(1-d-s)\Gamma(-s)}{\Gamma(e+s)}ds$$

$$=\frac{\Gamma(a)\Gamma(b)\Gamma(c)\Gamma(1-d+a)\Gamma(1-d+b)\Gamma(1-d+c)}{\Gamma(e-a)\Gamma(e-b)\Gamma(e-c)}$$

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 (Gasper & Rahman 2004, chapter 4).

References

Barnes, E.W. (1908). "A new development of the theory of the hypergeometric functions". Proc. London Math. Soc. s2–6: 141–177. doi:10.1112/plms/s2-6.1.141. JFM 39.0506.01.
Barnes, E.W. (1910). "A transformation of generalised hypergeometric series". Quarterly Journal of Mathematics 41: 136–140. JFM 41.0503.01.
Gasper, George; Rahman, Mizan (2004). Basic hypergeometric series. Encyclopedia of Mathematics and its Applications 96 (2nd ed.). Cambridge University Press. ISBN 978-0-521-83357-8. MR 2128719.