Barnes' Lemma

If a contour in the complex plane is curved such that it separates the increasing and decreasing sequences of poles, then

where  is the gamma function (Bailey 1935, p. 7).

Barnes' second lemma states that

provided that  (Bailey 1935, pp. 42-43).

REFERENCES:

Bailey, W. N. "Barnes' Lemma" and "Barnes' Second Lemma." §1.7 and 6.2 in Generalised Hypergeometric Series. Cambridge, England: University Press, pp. 7 and 42-43, 1935.

Barnes, E. W. "A New Development in the Theory of the Hypergeometric Functions." Proc. London Math. Soc. 6, 141-177, 1908.