Gauss hypergeometric function 2F1: Series representations (formula 07.23.06.0067)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Series representations

Residue representations

General case

http://functions.wolfram.com/07.23.06.0067.01

Input Form

Hypergeometric2F1[a, b, c, z] == (-((Csc[(a - b) Pi] Gamma[c])/(Gamma[a] Gamma[b]))) ((Sin[(a - c) Pi] (Sum[Residue[((Gamma[s] Gamma[1 + a - c - s])/ ((-(1/z))^s Gamma[1 + a - b - s])) Gamma[a - s], {s, a + j}], {j, 0, Infinity}] + Sum[Residue[((Gamma[s] Gamma[a - s])/ ((-(1/z))^s Gamma[1 + a - b - s])) Gamma[1 + a - c - s], {s, 1 + a - c + j}], {j, 0, Infinity}]))/(-z)^a + (Sin[(b - c) Pi] (Sum[Residue[((Gamma[s] Gamma[1 + b - c - s])/ ((-(1/z))^s Gamma[1 + b - a - s])) Gamma[b - s], {s, b + j}], {j, 0, Infinity}] + Sum[Residue[((Gamma[b - s] Gamma[s])/ ((-(1/z))^s Gamma[1 + b - a - s])) Gamma[1 + b - c - s], {s, 1 + b - c + j}], {j, 0, Infinity}]))/(-z)^b) /; Abs[z] < 1 && !Element[a - b, Integers] && !Element[c, Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", "b", ",", "c", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["Csc", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", "c", "]"]], " "]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]]]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "c"]], ")"]], " ", "\[Pi]"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "c", "-", "s"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "z"]]], ")"]], RowBox[List["-", "s"]]], " "]], RowBox[List[" ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "b", "-", "s"]], "]"]]]]], RowBox[List["Gamma", "[", RowBox[List["a", "-", "s"]], "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["a", "+", "j"]]]], "}"]]]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], RowBox[List["Gamma", "[", RowBox[List["a", "-", "s"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "z"]]], ")"]], RowBox[List["-", "s"]]], " "]], RowBox[List[" ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "b", "-", "s"]], "]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "c", "-", "s"]], "]"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["1", "+", "a", "-", "c", "+", "j"]]]], "}"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", "\[Pi]"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "b"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "b", "-", "c", "-", "s"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "z"]]], ")"]], RowBox[List["-", "s"]]], " "]], RowBox[List[" ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "b", "-", "a", "-", "s"]], "]"]]]]], RowBox[List["Gamma", "[", RowBox[List["b", "-", "s"]], "]"]]]], " ", ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["b", "+", "j"]]]], "}"]]]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["b", "-", "s"]], "]"]], RowBox[List["Gamma", "[", "s", "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "z"]]], ")"]], RowBox[List["-", "s"]]], " "]], RowBox[List[" ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "b", "-", "a", "-", "s"]], "]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "b", "-", "c", "-", "s"]], "]"]]]], " ", ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["1", "+", "b", "-", "c", "+", "j"]]]], "}"]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["a", "-", "b"]], ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["c", ",", "Integers"]], "]"]], "]"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ; </mo> <mi> c </mi> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> c </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mrow> <msub> <mi> res </mi> <mi> s </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> - </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mrow> <msub> <mi> res </mi> <mi> s </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> - </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mrow> <msub> <mi> res </mi> <mi> s </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mrow> <msub> <mi> res </mi> <mi> s </mi> </msub> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> - </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> <mo> + </mo> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> c </mi> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Hypergeometric2F1 </ci> <ci> a </ci> <ci> b </ci> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <csc /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <pi /> </apply> </apply> <apply> <ci> Gamma </ci> <ci> c </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <sin /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <pi /> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <ci> s </ci> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> s </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> a </ci> <ci> j </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <ci> s </ci> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> s </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <sin /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <pi /> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <ci> s </ci> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> s </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> b </ci> <ci> j </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <ci> s </ci> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <ci> s </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <notin /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <integers /> </apply> <apply> <notin /> <ci> c </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List["a_", ",", "b_", ",", "c_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Csc", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", "c", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "c"]], ")"]], " ", "\[Pi]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "c", "-", "s"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "z"]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "-", "s"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "b", "-", "s"]], "]"]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["a", "+", "j"]]]], "}"]]]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "-", "s"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "z"]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "c", "-", "s"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "b", "-", "s"]], "]"]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["1", "+", "a", "-", "c", "+", "j"]]]], "}"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List["b", "-", "c"]], ")"]], " ", "\[Pi]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "b", "-", "c", "-", "s"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "z"]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "-", "s"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "b", "-", "a", "-", "s"]], "]"]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["b", "+", "j"]]]], "}"]]]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List["Residue", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["b", "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", "s", "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "z"]]], ")"]], RowBox[List["-", "s"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "b", "-", "c", "-", "s"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "b", "-", "a", "-", "s"]], "]"]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List["1", "+", "b", "-", "c", "+", "j"]]]], "}"]]]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", "b", "]"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]], "&&", RowBox[List["!", RowBox[List[RowBox[List["a", "-", "b"]], "\[Element]", "Integers"]]]], "&&", RowBox[List["!", RowBox[List["c", "\[Element]", "Integers"]]]]]]]]]]]]

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

2007-05-02

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅,a₆},{b₁,b₂,b₃,b₄,b₅},z]
HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]