Barnes integral

Template:Short description 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 by

${}_2{F}_1(a,b;c;z)=\frac{\mathrm{\Gamma }(c)}{\mathrm{\Gamma }(a)\mathrm{\Gamma }(b)}\frac{1}{2\pi i}\underset{-i\mathrm{\infty }}{\overset{i\mathrm{\infty }}{\int }}\frac{\mathrm{\Gamma }(a+s)\mathrm{\Gamma }(b+s)\mathrm{\Gamma }(-s)}{\mathrm{\Gamma }(c+s)}(-z{)}^{s}ds,$

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 _pF_q in a similar way .

Barnes lemmas

The first Barnes lemma states

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

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 states

$\frac{1}{2\pi i}\underset{-i\mathrm{\infty }}{\overset{i\mathrm{\infty }}{\int }}\frac{\mathrm{\Gamma }(a+s)\mathrm{\Gamma }(b+s)\mathrm{\Gamma }(c+s)\mathrm{\Gamma }(1-d-s)\mathrm{\Gamma }(-s)}{\mathrm{\Gamma }(e+s)}ds$

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

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 .

References

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