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

[math]\displaystyle{ {}_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, }[/math]

see also (Andrews Askey). This equality can be obtained by moving the contour to the right while picking up the residues at s = 0, 1, 2, ... . for [math]\displaystyle{ z\ll 1 }[/math], 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

[math]\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)}. }[/math]

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

[math]\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 }[/math]

[math]\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)} }[/math]

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).

References

• Andrews, G.E.; Askey, R.; Roy, R. (1999). Special functions. Encyclopedia of Mathematics and its Applications. 71. Cambridge University Press. ISBN 0-521-62321-9. 
• 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. https://zenodo.org/record/1447796. 
• Barnes, E.W. (1910). "A transformation of generalised hypergeometric series". Quarterly Journal of Mathematics 41: 136–140. 
• 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. 
• Slater, Lucy Joan (1966). Generalized Hypergeometric Functions. Cambridge, UK: Cambridge University Press. ISBN 0-521-06483-X. https://archive.org/details/generalizedhyper0000unse.  (there is a 2008 paperback with ISBN:978-0-521-09061-2)