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Barnes' Lemma

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

1/(2pii)int_(-iinfty)^(iinfty)Gamma(alpha+s)Gamma(beta+s)Gamma(gamma-s)Gamma(delta-s)ds =(Gamma(alpha+gamma)Gamma(alpha+delta)Gamma(beta+gamma)Gamma(beta+delta))/(Gamma(alpha+beta+gamma+delta)),

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

Barnes' second lemma states that

int1/(2pii)(Gamma(alpha_1+s)Gamma(alpha_2+s)Gamma(alpha_3+s)Gamma(1-beta_1-s)Gamma(-s)ds)/(Gamma(beta_2+s)) =(Gamma(alpha_1)Gamma(alpha_2)Gamma(alpha_3)Gamma(1-beta_1+alpha_1)Gamma(1-beta_1+alpha_2)Gamma(1-beta_1+alpha_3))/(Gamma(beta_2-alpha_1)Gamma(beta_2-alpha_2)Gamma(beta_2-alpha_3))

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

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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.

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Barnes' Lemma

Cite this as:

Weisstein, Eric W. "Barnes' Lemma." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BarnesLemma.html

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