A type of integral containing gamma functions in
its integrand. A typical such integral is given by
where
is real, ,
,
,
and
are positive, and the contour is a straight line parallel
to the imaginary axis with indentations if necessary
to avoid poles of the integrand.
Barnes, E. W. "A New Development in the Theory of the Hypergeometric Functions." Proc. London Math. Soc.6, 141-177,
1908.Dixon, A. L. and Ferrar, W. L. "A Class of Discontinuous
Integrals." Quart. J. Math. (Oxford Ser.)7, 81-96, 1936.Erdélyi,
A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "Mellin-Barnes Integrals."
§1.19 in Higher
Transcendental Functions, Vol. 1. New York: Krieger, pp. 49-50,
1981.Mellin, H. "Om Definita Integraler." Acta Societatis
Scientiarum Fennicae20, No. 7, 1-39, 1895.Mellin, H.
"Abrißeiner einheitlichen Theorie der Gamma- und der hypergeometrischen
Funktionen." Math. Ann.68, 305-337, 1909.Paris,
R. B. and Kaminski, D. Asymptotics
and Mellin-Barnes Integrals. Cambridge, England: Cambridge University Press,
2001.Pincherle, S. Atti d. R. Academia dei Lincei, Ser. 4, Rendiconti4,
694-700 and 792-799, 1888.Ramanujan, S. Collected
Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar,
and B. M. Wilson). Providence, RI: Amer. Math. Soc., p. 216, 2000.Whittaker,
E. T. and Watson, G. N. A
Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University
Press, p. 289, 1990.
is real, 
,
,
,
and 
are positive, and the contour is a straight line parallel
to the imaginary axis with indentations if necessary
to avoid poles of the integrand.
