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In mathematics, a '''Barnes integral''' or '''[[Hjalmar Mellin|Mellin]]–Barnes integral''' is a [[contour integral]] involving a product of [[gamma function]]s. They were introduced by {{harvs|txt|authorlink=Ernest William Barnes|first=Ernest William |last=Barnes|year1=1908|year2=1910}}. They are closely related to [[generalized hypergeometric series]].
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The integral is usually taken along a contour which is a deformation of the imaginary axis passing to the left of all poles of factors of the form Γ(''a''&nbsp;+&nbsp;''s'') and to the right of all poles of factors of the form Γ(''a''&nbsp;&minus;&nbsp;''s'').
 
==Hypergeometric series==
The [[hypergeometric function]] is given as a Barnes integral {{harv|Barnes|1908}} by
 
:<math>{}_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>
 
This equality can be obtained by moving the contour to the right while picking up the [[Residue (complex analysis)|residues]] at ''s'' = 0, 1, 2, ... . Given proper convergence conditions, one can relate more general Barnes' integrals and [[generalized hypergeometric function]]s <sub>''p''</sub>''F''<sub>''q''</sub> in a similar way.
 
==Barnes lemmas==
The first Barnes lemma {{harv|Barnes|1908}} states
 
:<math>\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 <sub>2</sub>''F''<sub>1</sub> 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 {{harv|Barnes|1910}} states
 
:<math>\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>=\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''&nbsp;+&nbsp;''b''&nbsp;+&nbsp;''c''&nbsp;&minus;&nbsp;''d''&nbsp;+&nbsp;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 {{harv|Gasper|Rahman|2004|loc=chapter 4}}.
 
==References==
*{{cite journal | last1= Barnes | first1= E.W. | title= A new development of the theory of the hypergeometric functions | doi= 10.1112/plms/s2-6.1.141 | year= 1908 | journal= Proc. London Math. Soc. | volume= s2-6 | pages= 141–177 | jfm= 39.0506.01 | ref= harv}}
*{{cite journal | last1= Barnes | first1= E.W. | title= A transformation of generalised hypergeometric series | year= 1910 | journal= [[Quarterly Journal of Mathematics]] | volume= 41 | pages= 136–140 | jfm= 41.0503.01 | ref= harv}}
*{{cite book | last1= Gasper | first1= George | last2= Rahman | first2= Mizan | title= Basic hypergeometric series | publisher= [[Cambridge University Press]] | edition= 2nd | series= Encyclopedia of Mathematics and its Applications | isbn= 978-0-521-83357-8 | year= 2004 | volume= 96 | mr= 2128719 | ref= harv}}
 
[[Category:Special functions]]
[[Category:Hypergeometric functions]]

Latest revision as of 22:02, 3 December 2014

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