Barnes integral

{{Short description|Contour integral involving a product of gamma functions}}

In mathematics, a Barnes integral or Mellin–Barnes integral is a contour integral involving a product of gamma functions. 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.

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 {{harv|Barnes|1908}} by

: ${}_2F_1(a,b;c;z) =\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \frac{\Gamma(a+s)\Gamma(b+s)\Gamma(-s)}{\Gamma(c+s)}(-z)^s\,ds,$

see also {{harv|Andrews|Askey|Roy|1999|loc=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 $z\ll 1$ , 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 {{harv|Slater|1966}}.

Barnes lemmas

The first Barnes lemma {{harv|Barnes|1908}} states

: $\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \Gamma(a+s)\Gamma(b+s)\Gamma(c-s)\Gamma(d-s)ds
=\frac{\Gamma(a+c)\Gamma(a+d)\Gamma(b+c)\Gamma(b+d)}{\Gamma(a+b+c+d)}.$

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 {{harv|Barnes|1910}} states

: $\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} \frac{\Gamma(a+s)\Gamma(b+s)\Gamma(c+s)\Gamma(1-d-s)\Gamma(-s)}{\Gamma(e+s)}ds$

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

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 {{harv|Gasper|Rahman|2004|loc=chapter 4}}.

References

{{cite book | last1= Andrews | first1= G.E. | authorlink1= George Andrews (mathematician)| last2= Askey | first2= R. | authorlink2= Richard Askey | last3= Roy | first3= R. | title= Special functions | publisher= Cambridge University Press | series= Encyclopedia of Mathematics and its Applications | volume= 71 | year= 1999 | isbn= 0-521-62321-9 | mr= 1688958 }}
{{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 | url= https://zenodo.org/record/1447796 | doi-access= free }}
{{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 }}
{{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 }}
{{cite book | last= Slater | first= Lucy Joan | author-link= Lucy Joan Slater | title= Generalized Hypergeometric Functions | url= https://archive.org/details/generalizedhyper0000unse | url-access= registration | publisher= Cambridge University Press | location= Cambridge, UK | year= 1966 | isbn= 0-521-06483-X | mr= 0201688 | zbl= 0135.28101 }} (there is a 2008 paperback with {{ISBN|978-0-521-09061-2}})

Category:Special functions

Category:Hypergeometric functions