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variants of this functions
Gamma






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Gamma[z] > Integration > Definite integration > Involving the direct function





http://functions.wolfram.com/06.05.21.0001.01









  


  










Input Form





Integrate[Gamma[a + t] Gamma[b + t] Gamma[c - t] Gamma[d - t], {t, \[Gamma] - I Infinity, \[Gamma] + I Infinity}] == 2 Pi I ((Gamma[a + c] Gamma[a + d] Gamma[b + c] Gamma[b + d])/ Gamma[a + b + c + d]) /; -Min[Re[a], Re[b]] < \[Gamma] < Min[Re[c], Re[d]]










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> &#8747; </mo> <mrow> <mi> &#947; </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> <mrow> <mi> &#947; </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#8734; </mi> </mrow> </mrow> </msubsup> <mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> c </mi> <mo> + </mo> <mi> d </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> min </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &lt; </mo> <mi> &#947; </mi> <mo> &lt; </mo> <mrow> <mi> min </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> c </mi> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <apply> <plus /> <ci> &#947; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <infinity /> </apply> </apply> </apply> </lowlimit> <uplimit> <apply> <plus /> <ci> &#947; </ci> <apply> <times /> <imaginaryi /> <infinity /> </apply> </apply> </uplimit> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> t </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <ci> t </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> d </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> c </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> d </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <ci> c </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <ci> d </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> <ci> c </ci> <ci> d </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <lt /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <min /> <apply> <real /> <ci> a </ci> </apply> <apply> <real /> <ci> b </ci> </apply> </apply> </apply> <ci> &#947; </ci> <apply> <min /> <apply> <real /> <ci> c </ci> </apply> <apply> <real /> <ci> d </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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Date Added to functions.wolfram.com (modification date)





2001-10-29