Barnes' Lemma

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


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

Barnes' second lemma states that


provided that beta_1+beta_2=alpha_1+alpha_2+alpha_3+1 (Bailey 1935, pp. 42-43).

Explore with Wolfram|Alpha


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.

Referenced on Wolfram|Alpha

Barnes' Lemma

Cite this as:

Weisstein, Eric W. "Barnes' Lemma." From MathWorld--A Wolfram Web Resource.

Subject classifications