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 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).

Hypergeometric series

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).

Barnes lemmas

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.

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

• 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)