巴恩斯积分

巴恩斯积分（英語：），由英國數學家歐內斯特·巴恩斯所推導而得，涉及Γ函数乘積的周回積分運算，研究複分析的工具，因芬蘭數學家亞爾馬·梅林的部份貢獻，又稱「梅林－巴恩斯积分」，與广义超几何函数高度相關。

Barnes Integral 3D

Barnes Integral density plot

定义如下：[1]

B(\sigma ,\mu )={\frac {\Gamma (\sigma /2+1)*\Gamma (\sigma /2+1)}{2^{\mu +1}*\Gamma (\sigma /2-v/2+\mu /2+1)*\Gamma (\sigma /2+v/2+\mu /2+3/2)}}

超几何函数

已知超几何函数時，巴恩斯积分如下而得[2]

{}_{2}F_{1}(a,b;c;z)={\frac {\Gamma (c)}{\Gamma (a)\Gamma (b)}}{\frac {1}{2\pi i}}\int _{-i\infty }^{i\infty }{\frac {\Gamma (a+s)\Gamma (b+s)\Gamma (-s)}{\Gamma (c+s)}}(-z)^{s}\,ds.

Frank Oliver NIST Handbook of Mathematical Functions, p371, Cambridge University Press, 2010
（Barnes 1908）

Barnes, E.W. . Proc. London Math. Soc. 1908, s2–6: 141–177. JFM 39.0506.01. doi:10.1112/plms/s2-6.1.141.
Barnes, E.W. . Quarterly Journal of Mathematics. 1910, 41: 136–140. JFM 41.0503.01.
Gasper, George; Rahman, Mizan. . Encyclopedia of Mathematics and its Applications 96 2nd. Cambridge University Press. 2004. ISBN 978-0-521-83357-8. MR 2128719.

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