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In mathematics, a Barnes integral or
Mellin Mellin is a village and a former municipality in the district Altmarkkreis Salzwedel, in Saxony-Anhalt, Germany. Since 1 January 2009, it is part of the municipality Beetzendorf Beetzendorf is a municipality in the district Altmarkkreis Salzwe ...
–Barnes integral is a
contour integral In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. ...
involving a product of
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
s. They were introduced by . They are closely related to
generalized hypergeometric series In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which ...
. 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 :_2F_1(a,b;c;z) =\frac \frac \int_^ \frac(-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 function In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which ...
s ''p''''F''''q'' in a similar way .


Barnes lemmas

The first Barnes lemma states :\frac \int_^ \Gamma(a+s)\Gamma(b+s)\Gamma(c-s)\Gamma(d-s)ds =\frac. This is an analogue of Gauss's 2''F''1 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 \int_^ \fracds :=\frac 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 In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called h ...
, and many of the other results can also be extended to this case .


References

* * * * * (there is a 2008 paperback with {{ISBN, 978-0-521-09061-2) Special functions Hypergeometric functions