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).
Contents
Hypergeometric series
The hypergeometric function is given as a Barnes integral (Barnes 1908) by
2
F
1
(
a
,
b
;
c
;
z
)
=
Γ
(
c
)
Γ
(
a
)
Γ
(
b
)
1
2
π
i
∫
−
i
∞
i
∞
Γ
(
a
+
s
)
Γ
(
b
+
s
)
Γ
(
−
s
)
Γ
(
c
+
s
)
(
−
z
)
s
d
s
,
{\displaystyle {}_{2}F_{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 (Andrews, Askey & Roy 1999, 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
≪
1
{\displaystyle z\ll 1}
, and by analytic continuation elsewhere. Given proper convergence conditions, one can relate more general Barnes' integrals and generalized hypergeometric functions pFq in a similar way (Slater 1966).
Barnes lemmas
The first Barnes lemma (Barnes 1908) states
1
2
π
i
∫
−
i
∞
i
∞
Γ
(
a
+
s
)
Γ
(
b
+
s
)
Γ
(
c
−
s
)
Γ
(
d
−
s
)
d
s
=
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 2004, chapter 4).