Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Gamma






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Gamma[z] > Operations > Residues of ratios of gamma functions > Case of simple poles





http://functions.wolfram.com/06.05.25.0006.01









  


  










Input Form





Residue[(Product[Gamma[Subscript[\[GothicCapitalA], k] s + Subscript[a, k]], {k, 1, \[ScriptCapitalA]}] Product[Gamma[Subscript[b, k] - Subscript[\[GothicCapitalB], k] s], {k, 1, \[ScriptCapitalB]}])/ (Product[Gamma[Subscript[\[GothicCapitalC], k] s + Subscript[c, k]], {k, 1, \[ScriptCapitalC]}] Product[Gamma[Subscript[d, k] - Subscript[\[GothicCapitalD], k] s], {k, 1, \[ScriptCapitalD]}])/z^s, {s, -((Subscript[a, 1] + l)/Subscript[\[GothicCapitalA], 1])}] == (((-1)^l Product[Gamma[Subscript[a, k] - Subscript[\[GothicCapitalA], k] ((Subscript[a, 1] + l)/Subscript[\[GothicCapitalA], 1])] Product[Gamma[Subscript[b, k] + Subscript[\[GothicCapitalB], k] ((Subscript[a, 1] + l)/Subscript[\[GothicCapitalA], 1])], {k, 1, B}], {k, 2, A}])/(Subscript[\[GothicCapitalA], 1] l! Product[Gamma[Subscript[c, k] - Subscript[\[GothicCapitalC], k] ((Subscript[a, 1] + l)/Subscript[\[GothicCapitalA], 1])] Product[Gamma[Subscript[d, k] + Subscript[\[GothicCapitalD], k] ((Subscript[a, 1] + l)/Subscript[\[GothicCapitalA], 1])], {k, 1, \[ScriptCapitalD]}], {k, 1, \[ScriptCapitalC]}])) z^((Subscript[a, 1] + l)/Subscript[\[GothicCapitalA], 1]) /; Element[l, Integers] && l >= 0 && !Element[Subscript[a, j] - Subscript[\[GothicCapitalA], j] ((Subscript[a, 1] + l)/Subscript[\[GothicCapitalA], 1]), Integers] && 2 <= j <= \[ScriptCapitalA] && !(Element[-Subscript[b, j] - Subscript[\[GothicCapitalB], j] ((Subscript[a, 1] + l)/Subscript[\[GothicCapitalA], 1]), Integers] && -Subscript[b, j] - Subscript[\[GothicCapitalB], j] ((Subscript[a, 1] + l)/Subscript[\[GothicCapitalA], 1]) >= 0) && 1 <= j <= \[ScriptCapitalB] && !(Element[Subscript[c, j] - Subscript[\[GothicCapitalC], j] ((Subscript[a, 1] + l)/Subscript[\[GothicCapitalA], 1]), Integers] && Subscript[c, j] - Subscript[\[GothicCapitalC], j] ((Subscript[a, 1] + l)/ Subscript[\[GothicCapitalA], 1]) >= 0) && 1 <= j <= \[ScriptCapitalC] && !(Element[-Subscript[d, j] - Subscript[\[GothicCapitalD], j] ((Subscript[a, 1] + l)/Subscript[\[GothicCapitalA], 1]), Integers] && -Subscript[d, j] - Subscript[\[GothicCapitalD], j] ((Subscript[a, 1] + l)/Subscript[\[GothicCapitalA], 1]) >= 0) && 1 <= j <= \[ScriptCapitalD]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[ScriptCapitalA]"], RowBox[List["Gamma", "[", RowBox[List[RowBox[List[SubscriptBox["\[GothicCapitalA]", "k"], "s"]], "+", SubscriptBox["a", "k"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[ScriptCapitalB]"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "k"], "-", RowBox[List[SubscriptBox["\[GothicCapitalB]", "k"], "s"]]]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[ScriptCapitalC]"], RowBox[List["Gamma", "[", RowBox[List[RowBox[List[SubscriptBox["\[GothicCapitalC]", "k"], "s"]], "+", SubscriptBox["c", "k"]]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[ScriptCapitalD]"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["d", "k"], "-", RowBox[List[SubscriptBox["\[GothicCapitalD]", "k"], "s"]]]], "]"]]]]]]], SuperscriptBox["z", RowBox[List["-", "s"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["-", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "l"]], SubscriptBox["\[GothicCapitalA]", "1"]]]]]], "}"]]]], "]"]], " ", "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "l"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "2"]], "A"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "k"], "-", RowBox[List[SubscriptBox["\[GothicCapitalA]", "k"], FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "l"]], SubscriptBox["\[GothicCapitalA]", "1"]]]]]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "B"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "k"], "+", RowBox[List[SubscriptBox["\[GothicCapitalB]", "k"], FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "l"]], SubscriptBox["\[GothicCapitalA]", "1"]]]]]], "]"]]]]]]]]]], RowBox[List[SubscriptBox["\[GothicCapitalA]", "1"], RowBox[List["l", "!"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[ScriptCapitalC]"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["c", "k"], "-", RowBox[List[SubscriptBox["\[GothicCapitalC]", "k"], FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "l"]], SubscriptBox["\[GothicCapitalA]", "1"]]]]]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[ScriptCapitalD]"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["d", "k"], "+", RowBox[List[SubscriptBox["\[GothicCapitalD]", "k"], FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "l"]], SubscriptBox["\[GothicCapitalA]", "1"]]]]]], "]"]]]]]]]]]]], SuperscriptBox["z", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "l"]], SubscriptBox["\[GothicCapitalA]", "1"]]]]]]], "/;", RowBox[List[RowBox[List["l", "\[Element]", "Integers"]], "\[And]", RowBox[List["l", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", RowBox[List[SubscriptBox["\[GothicCapitalA]", "j"], FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "l"]], SubscriptBox["\[GothicCapitalA]", "1"]]]]]], "\[Element]", "Integers"]], "]"]], "\[And]", RowBox[List["2", "\[LessEqual]", "j", "\[LessEqual]", "\[ScriptCapitalA]"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["b", "j"]]], "-", RowBox[List[SubscriptBox["\[GothicCapitalB]", "j"], FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "l"]], SubscriptBox["\[GothicCapitalA]", "1"]]]]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["b", "j"]]], "-", RowBox[List[SubscriptBox["\[GothicCapitalB]", "j"], FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "l"]], SubscriptBox["\[GothicCapitalA]", "1"]]]]]], "\[GreaterEqual]", "0"]]]], "]"]], "\[And]", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "\[ScriptCapitalB]"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["c", "j"], "-", RowBox[List[SubscriptBox["\[GothicCapitalC]", "j"], FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "l"]], SubscriptBox["\[GothicCapitalA]", "1"]]]]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["c", "j"], "-", RowBox[List[SubscriptBox["\[GothicCapitalC]", "j"], FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "l"]], SubscriptBox["\[GothicCapitalA]", "1"]]]]]], "\[GreaterEqual]", "0"]]]], "]"]], "\[And]", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "\[ScriptCapitalC]"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["d", "j"]]], "-", RowBox[List[SubscriptBox["\[GothicCapitalD]", "j"], FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "l"]], SubscriptBox["\[GothicCapitalA]", "1"]]]]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["d", "j"]]], "-", RowBox[List[SubscriptBox["\[GothicCapitalD]", "j"], FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "l"]], SubscriptBox["\[GothicCapitalA]", "1"]]]]]], "\[GreaterEqual]", "0"]]]], "]"]], "\[And]", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "\[ScriptCapitalD]"]]]]]]]]










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> <mrow> <msub> <mi> res </mi> <mi> s </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#119964; </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> &#120068; </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <mi> s </mi> </mrow> <mo> + </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8492; </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> - </mo> <mrow> <msub> <mi> &#120069; </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <mi> s </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> C </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <msub> <mi> &#8493; </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <mi> s </mi> </mrow> <mo> + </mo> <msub> <mi> c </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#119967; </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> - </mo> <mrow> <msub> <mi> &#120071; </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <mi> s </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> l </mi> </mrow> <msub> <mi> &#120068; </mi> <mn> 1 </mn> </msub> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> l </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mi> &#119964; </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> - </mo> <mrow> <msub> <mi> &#120068; </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> l </mi> </mrow> <msub> <mi> &#120068; </mi> <mn> 1 </mn> </msub> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8492; </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> + </mo> <mrow> <msub> <mi> &#120069; </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> l </mi> </mrow> <msub> <mi> &#120068; </mi> <mn> 1 </mn> </msub> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msub> <mi> &#120068; </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mrow> <mi> l </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> C </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> - </mo> <mrow> <msub> <mi> &#8493; </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> l </mi> </mrow> <msub> <mi> &#120068; </mi> <mn> 1 </mn> </msub> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#119967; </mi> </munderover> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> + </mo> <mrow> <msub> <mi> &#120071; </mi> <mi> k </mi> </msub> <mo> &#8290; </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> l </mi> </mrow> <msub> <mi> &#120068; </mi> <mn> 1 </mn> </msub> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> l </mi> </mrow> <msub> <mi> &#120068; </mi> <mn> 1 </mn> </msub> </mfrac> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> l </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> - </mo> <mrow> <msub> <mi> &#120068; </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> l </mi> </mrow> <msub> <mi> &#120068; </mi> <mn> 1 </mn> </msub> </mfrac> </mrow> </mrow> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mn> 2 </mn> <mo> &#8804; </mo> <mi> j </mi> <mo> &#8804; </mo> <mi> &#119964; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> - </mo> <mrow> <msub> <mi> &#120069; </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> l </mi> </mrow> <msub> <mi> &#120068; </mi> <mn> 1 </mn> </msub> </mfrac> </mrow> </mrow> <mo> &#8713; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> j </mi> <mo> &#8804; </mo> <mi> &#8492; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msub> <mi> c </mi> <mi> j </mi> </msub> <mo> - </mo> <mrow> <msub> <mi> &#8493; </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> l </mi> </mrow> <msub> <mi> &#120068; </mi> <mn> 1 </mn> </msub> </mfrac> </mrow> </mrow> <mo> &#8713; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> j </mi> <mo> &#8804; </mo> <mi> &#119966; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msub> <mi> d </mi> <mi> j </mi> </msub> </mrow> <mo> - </mo> <mrow> <msub> <mi> &#120071; </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <mfrac> <mrow> <mtext> </mtext> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mi> l </mi> </mrow> </mrow> <msub> <mi> &#120068; </mi> <mn> 1 </mn> </msub> </mfrac> </mrow> </mrow> <mo> &#8713; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mn> 1 </mn> <mo> &#8804; </mo> <mi> j </mi> <mo> &#8804; </mo> <mi> &#119967; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <ci> s </ci> </apply> <apply> <times /> <apply> <times /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> &#119964; </ci> </uplimit> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#120068; </ci> <ci> k </ci> </apply> <ci> s </ci> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> &#8492; </ci> </uplimit> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#120069; </ci> <ci> k </ci> </apply> <ci> s </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> C </ci> </uplimit> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#8493; </ci> <ci> k </ci> </apply> <ci> s </ci> </apply> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> &#119967; </ci> </uplimit> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#120071; </ci> <ci> k </ci> </apply> <ci> s </ci> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> l </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#120068; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> l </ci> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 2 </cn> </lowlimit> <uplimit> <ci> &#119964; </ci> </uplimit> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#120068; </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> l </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#120068; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> &#8492; </ci> </uplimit> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#120069; </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> l </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#120068; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#120068; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <factorial /> <ci> l </ci> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> C </ci> </uplimit> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#8493; </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> l </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#120068; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> &#119967; </ci> </uplimit> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#120071; </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> l </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#120068; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> l </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#120068; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> l </ci> <integers /> </apply> <apply> <notin /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#120068; </ci> <ci> j </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> l </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#120068; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <integers /> </apply> <apply> <leq /> <cn type='integer'> 2 </cn> <ci> j </ci> <ci> &#119964; </ci> </apply> <apply> <notin /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> j </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#120069; </ci> <ci> j </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> l </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#120068; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <integers /> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> j </ci> <ci> &#8492; </ci> </apply> <apply> <notin /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#8493; </ci> <ci> j </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> l </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#120068; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <integers /> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> j </ci> <ci> &#119966; </ci> </apply> <apply> <notin /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> j </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#120071; </ci> <ci> j </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> l </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#120068; </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <integers /> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> j </ci> <ci> &#119967; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k_", "=", "1"]], "\[ScriptCapitalA]_"], RowBox[List["Gamma", "[", RowBox[List[RowBox[List[SubscriptBox["\[GothicCapitalA]_", "k_"], " ", "s_"]], "+", SubscriptBox["a_", "k_"]]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k_", "=", "1"]], "\[ScriptCapitalB]_"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b_", "k_"], "-", RowBox[List[SubscriptBox["\[GothicCapitalB]_", "k_"], " ", "s_"]]]], "]"]]]]]], ")"]], " ", SuperscriptBox["z_", RowBox[List["-", "s_"]]]]], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k_", "=", "1"]], "\[ScriptCapitalC]_"], RowBox[List["Gamma", "[", RowBox[List[RowBox[List[SubscriptBox["\[GothicCapitalC]_", "k_"], " ", "s_"]], "+", SubscriptBox["c_", "k_"]]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k_", "=", "1"]], "\[ScriptCapitalD]_"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["d_", "k_"], "-", RowBox[List[SubscriptBox["\[GothicCapitalD]_", "k_"], " ", "s_"]]]], "]"]]]]]]], ",", RowBox[List["{", RowBox[List["s_", ",", RowBox[List["-", FractionBox[RowBox[List[SubscriptBox["a_", "1"], "+", "l_"]], SubscriptBox["\[GothicCapitalA]_", "1"]]]]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "l"], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "2"]], "A"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["aa", "k"], "-", FractionBox[RowBox[List[SubscriptBox["\[GothicCapitalA]\[GothicCapitalA]", "k"], " ", RowBox[List["(", RowBox[List[SubscriptBox["aa", "1"], "+", "l"]], ")"]]]], SubscriptBox["\[GothicCapitalA]\[GothicCapitalA]", "1"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "B"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["bb", "k"], "+", FractionBox[RowBox[List[SubscriptBox["\[GothicCapitalB]\[GothicCapitalB]", "k"], " ", RowBox[List["(", RowBox[List[SubscriptBox["aa", "1"], "+", "l"]], ")"]]]], SubscriptBox["\[GothicCapitalA]\[GothicCapitalA]", "1"]]]], "]"]]]]]]]]]], ")"]], " ", SuperscriptBox["z", FractionBox[RowBox[List[SubscriptBox["aa", "1"], "+", "l"]], SubscriptBox["\[GothicCapitalA]\[GothicCapitalA]", "1"]]]]], RowBox[List[SubscriptBox["\[GothicCapitalA]\[GothicCapitalA]", "1"], " ", RowBox[List["l", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[ScriptCapitalC]"], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["cc", "k"], "-", FractionBox[RowBox[List[SubscriptBox["\[GothicCapitalC]\[GothicCapitalC]", "k"], " ", RowBox[List["(", RowBox[List[SubscriptBox["aa", "1"], "+", "l"]], ")"]]]], SubscriptBox["\[GothicCapitalA]\[GothicCapitalA]", "1"]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[ScriptCapitalD]"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["dd", "k"], "+", FractionBox[RowBox[List[SubscriptBox["\[GothicCapitalD]\[GothicCapitalD]", "k"], " ", RowBox[List["(", RowBox[List[SubscriptBox["aa", "1"], "+", "l"]], ")"]]]], SubscriptBox["\[GothicCapitalA]\[GothicCapitalA]", "1"]]]], "]"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["l", "\[Element]", "Integers"]], "&&", RowBox[List["l", "\[GreaterEqual]", "0"]], "&&", RowBox[List["!", RowBox[List[RowBox[List[SubscriptBox["a", "j"], "-", FractionBox[RowBox[List[SubscriptBox["\[GothicCapitalA]", "j"], " ", RowBox[List["(", RowBox[List[SubscriptBox["aa", "1"], "+", "l"]], ")"]]]], SubscriptBox["\[GothicCapitalA]\[GothicCapitalA]", "1"]]]], "\[Element]", "Integers"]]]], "&&", RowBox[List["2", "\[LessEqual]", "j", "\[LessEqual]", "\[ScriptCapitalA]"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["b", "j"]]], "-", FractionBox[RowBox[List[SubscriptBox["\[GothicCapitalB]", "j"], " ", RowBox[List["(", RowBox[List[SubscriptBox["aa", "1"], "+", "l"]], ")"]]]], SubscriptBox["\[GothicCapitalA]\[GothicCapitalA]", "1"]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["b", "j"]]], "-", FractionBox[RowBox[List[SubscriptBox["\[GothicCapitalB]", "j"], " ", RowBox[List["(", RowBox[List[SubscriptBox["aa", "1"], "+", "l"]], ")"]]]], SubscriptBox["\[GothicCapitalA]\[GothicCapitalA]", "1"]]]], "\[GreaterEqual]", "0"]]]], ")"]]]], "&&", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "\[ScriptCapitalB]"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["c", "j"], "-", FractionBox[RowBox[List[SubscriptBox["\[GothicCapitalC]", "j"], " ", RowBox[List["(", RowBox[List[SubscriptBox["aa", "1"], "+", "l"]], ")"]]]], SubscriptBox["\[GothicCapitalA]\[GothicCapitalA]", "1"]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[SubscriptBox["c", "j"], "-", FractionBox[RowBox[List[SubscriptBox["\[GothicCapitalC]", "j"], " ", RowBox[List["(", RowBox[List[SubscriptBox["aa", "1"], "+", "l"]], ")"]]]], SubscriptBox["\[GothicCapitalA]\[GothicCapitalA]", "1"]]]], "\[GreaterEqual]", "0"]]]], ")"]]]], "&&", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "\[ScriptCapitalC]"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["d", "j"]]], "-", FractionBox[RowBox[List[SubscriptBox["\[GothicCapitalD]", "j"], " ", RowBox[List["(", RowBox[List[SubscriptBox["aa", "1"], "+", "l"]], ")"]]]], SubscriptBox["\[GothicCapitalA]\[GothicCapitalA]", "1"]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["d", "j"]]], "-", FractionBox[RowBox[List[SubscriptBox["\[GothicCapitalD]", "j"], " ", RowBox[List["(", RowBox[List[SubscriptBox["aa", "1"], "+", "l"]], ")"]]]], SubscriptBox["\[GothicCapitalA]\[GothicCapitalA]", "1"]]]], "\[GreaterEqual]", "0"]]]], ")"]]]], "&&", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "\[ScriptCapitalD]"]]]]]]]]]]










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





2001-10-29





© 1998-2014 Wolfram Research, Inc.