查看“︁巴恩斯积分”︁的源代码

[[File:Barnes Integral animation 01.gif|thumb|Barnes  Integral  3D]]
[[File:Barnes Integral animation 02.gif|thumb|Barnes Integral density plot]]
'''巴恩斯积分'''（{{lang-en|Barnes integral}}），由[[英國]]數學家{{tsl|en|Ernest Barnes|歐內斯特·巴恩斯}}所推導而得，涉及[[Γ函数]]乘積的{{tsl|ja|周回積分|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。

定义如下：<ref>Frank  Oliver  NIST  Handbook of Mathematical Functions, p371, Cambridge University Press, 2010</ref>

:<math>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)}</math>

==超几何函数==
已知[[超几何函数]]時，巴恩斯积分如下而得<ref>{{harv|Barnes|1908}}</ref>
:<math>{}_2F_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.</math>

==参考文献==
<references/>
*{{cite journal | last1= Barnes | first1= E.W. | title= A new development of the theory of the hypergeometric functions | doi= 10.1112/plms/s2-6.1.141 | year= 1908 | journal= Proc. London Math. Soc. | volume= s2-6 | pages= 141–177 | jfm= 39.0506.01 | ref= harv}}
*{{cite journal | last1= Barnes | first1= E.W. | title= A transformation of generalised hypergeometric series | year= 1910 | journal= [[Quarterly Journal of Mathematics]] | volume= 41 | pages= 136–140 | jfm= 41.0503.01 | ref= harv}}
*{{cite book | last1= Gasper | first1= George | last2= Rahman | first2= Mizan | title= Basic hypergeometric series | publisher= [[Cambridge University Press]] | edition= 2nd | series= Encyclopedia of Mathematics and its Applications | isbn= 978-0-521-83357-8 | year= 2004 | volume= 96 | mr= 2128719 | ref= harv}}

[[Category:特殊函数]]
[[Category:超幾何函數]]