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Mellin–Barnes integral : ウィキペディア英語版
Barnes integral
In mathematics, a Barnes integral or Mellin–Barnes integral is a contour integral involving a product of gamma functions. They were introduced by . 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 left of all poles of factors of the form Γ(''a'' + ''s'') and to the right of all poles of factors of the form Γ(''a'' − ''s'').
==Hypergeometric series==
The hypergeometric function is given as a Barnes integral by
: \frac \int_^ \frac(-z)^s\,ds.
This equality can be obtained by moving the contour to the right while picking up the residues at ''s'' = 0, 1, 2, ... . Given proper convergence conditions, one can relate more general Barnes' integrals and generalized hypergeometric functions ''p''''F''''q'' in a similar way.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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