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 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).

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

${\displaystyle {}_{2}F_{1}(a,b;c;z)={\frac {\Gamma (c)}{\Gamma (a)\Gamma (b)}}{\frac {1}{2\pi i}}\int _{-i\infty }^{i\infty }{\frac {\Gamma (a+s)\Gamma (b+s)\Gamma (-s)}{\Gamma (c+s)}}(-z)^{s}\,ds,}$

see also (Andrews, Askey & Roy 1999, Theorem 2.4.1). This equality can be obtained by moving the contour to the right while picking up the residues at s = 0, 1, 2, ... . for ${\displaystyle z\ll 1}$, and by analytic continuation elsewhere. Given proper convergence conditions, one can relate more general Barnes' integrals and generalized hypergeometric functions _pF_q in a similar way (Slater 1966).

The first Barnes lemma (Barnes 1908) states

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

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

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

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

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

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).

• Andrews, G.E.; Askey, R.; Roy, R. (1999). Special functions. Encyclopedia of Mathematics and its Applications. Vol. 71. Cambridge University Press. ISBN 0-521-62321-9. MR 1688958.
• 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. Vol. 96 (2nd ed.). Cambridge University Press. ISBN 978-0-521-83357-8. MR 2128719.
• Slater, Lucy Joan (1966). Generalized Hypergeometric Functions. Cambridge, UK: Cambridge University Press. ISBN 0-521-06483-X. MR 0201688. Zbl 0135.28101. (there is a 2008 paperback with ISBN 978-0-521-09061-2)