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variants of this functions
Gamma




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/06.05.25.0011.01









  


  








Input Form





Residue[(Product[Gamma[s + Subscript[a, k]], {k, 1, \[ScriptCapitalA]}] Product[Gamma[Subscript[b, k] - s], {k, 1, \[ScriptCapitalB]}])/ (Product[Gamma[s + Subscript[c, k]], {k, 1, \[ScriptCapitalC]}] Product[Gamma[Subscript[d, k] - s], {k, 1, \[ScriptCapitalD]}])/z^s, {s, Subscript[b, 3] + l}] == GammaResidue[ {{Subscript[a, 1], \[Ellipsis], Subscript[a, \[ScriptCapitalA]]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, \[ScriptCapitalB]]}}, {{Subscript[c, 1], \[Ellipsis], Subscript[c, \[ScriptCapitalC]]}, {Subscript[d, 1], \[Ellipsis], Subscript[d, \[ScriptCapitalD]]}}, {Subscript[b, 3], 3, l}, z] /; Element[Subscript[b, 2] - Subscript[b, 1], Integers] && Subscript[b, 2] - Subscript[b, 1] >= 0 && Element[Subscript[b, 3] - Subscript[b, 2], Integers] && Subscript[b, 3] - Subscript[b, 2] >= 0 && Element[l, Integers] && l >= 0 && !Element[Subscript[b, j] - Subscript[b, 1], Integers] && 4 <= j <= \[ScriptCapitalB] && !(Element[-Subscript[a, j] - Subscript[b, 1], Integers] && -Subscript[a, j] - Subscript[b, 1] >= 0) && 1 <= j <= \[ScriptCapitalA] && !(Element[Subscript[d, j] - Subscript[b, 3] - l, Integers] && Subscript[d, j] - Subscript[b, 3] - l >= 0) && 1 <= j <= \[ScriptCapitalD] && !(Element[-Subscript[c, j] - Subscript[b, 3] - l, Integers] && -Subscript[c, j] - Subscript[b, 3] - l >= 0) && 1 <= j <= \[ScriptCapitalC]








Standard Form





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MathML Form







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</ms> <ms> j </ms> <ms> &#8804; </ms> <ms> &#8492; </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> j </ms> </apply> </list> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> &#8713; </ms> <apply> <ci> TagBox </ci> <ms> &#8469; </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> &#8804; </ms> <ms> j </ms> <ms> &#8804; </ms> <ms> &#119964; </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 3 </ms> </apply> <ms> - </ms> <ms> l </ms> </list> </apply> <ms> &#8713; </ms> <apply> <ci> TagBox </ci> <ms> &#8469; </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> &#8804; </ms> <ms> j </ms> <ms> &#8804; </ms> <ms> &#119967; </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> c </ms> <ms> j </ms> </apply> </list> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 3 </ms> </apply> <ms> - </ms> <ms> l </ms> </list> </apply> <ms> &#8713; </ms> <apply> <ci> TagBox </ci> <ms> &#8469; </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> &#8804; </ms> <ms> j </ms> <ms> &#8804; </ms> <ms> &#119966; </ms> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </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["s_", "+", SubscriptBox["a_", "k_"]]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k_", "=", "1"]], "\[ScriptCapitalB]_"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b_", "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["s_", "+", SubscriptBox["c_", "k_"]]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k_", "=", "1"]], "\[ScriptCapitalD]_"], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["d_", "k_"], "-", "s_"]], "]"]]]]]]], ",", RowBox[List["{", RowBox[List["s_", ",", RowBox[List[SubscriptBox["b_", "3"], "+", "l_"]]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["GammaResidue", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["aa", "\[ScriptCapitalA]"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["bb", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["bb", "\[ScriptCapitalB]"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["cc", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["cc", "\[ScriptCapitalC]"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["dd", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["dd", "\[ScriptCapitalD]"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["bb", "3"], ",", "3", ",", "l"]], "}"]], ",", "z"]], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["bb", "1"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["bb", "1"]]], "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List[SubscriptBox["bb", "3"], "-", SubscriptBox["b", "2"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[SubscriptBox["bb", "3"], "-", SubscriptBox["b", "2"]]], "\[GreaterEqual]", "0"]], "&&", RowBox[List["l", "\[Element]", "Integers"]], "&&", RowBox[List["l", "\[GreaterEqual]", "0"]], "&&", RowBox[List["!", RowBox[List[RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["bb", "1"]]], "\[Element]", "Integers"]]]], "&&", RowBox[List["4", "\[LessEqual]", "j", "\[LessEqual]", "\[ScriptCapitalB]"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["a", "j"]]], "-", SubscriptBox["bb", "1"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["a", "j"]]], "-", SubscriptBox["bb", "1"]]], "\[GreaterEqual]", "0"]]]], ")"]]]], "&&", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "\[ScriptCapitalA]"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["d", "j"], "-", SubscriptBox["bb", "3"], "-", "l"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[SubscriptBox["d", "j"], "-", SubscriptBox["bb", "3"], "-", "l"]], "\[GreaterEqual]", "0"]]]], ")"]]]], "&&", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "\[ScriptCapitalD]"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["c", "j"]]], "-", SubscriptBox["bb", "3"], "-", "l"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["c", "j"]]], "-", SubscriptBox["bb", "3"], "-", "l"]], "\[GreaterEqual]", "0"]]]], ")"]]]], "&&", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", "\[ScriptCapitalC]"]]]]]]]]]]


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





2001-10-29