html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 HypergeometricPFQ

 http://functions.wolfram.com/07.26.06.0012.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 1]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] \[Proportional] ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[b, 3]])/ (Sqrt[Pi] Gamma[Subscript[a, 1]]^2)) (-z)^\[Chi] (Cos[2 Sqrt[-z] + Pi \[Chi]] (1 + O[1/z]) + (1/(16 Sqrt[-z])) (16 (Subscript[b, 1] Subscript[b, 2] + Subscript[b, 1] Subscript[b, 3] + Subscript[b, 2] Subscript[b, 3] - Subscript[a, 1]^2) + 2 (4 \[Chi] - 1) (6 Subscript[a, 1] + Subscript[b, 1] + Subscript[b, 2] + Subscript[b, 3] - 2) - 3) Sin[2 Sqrt[-z] + Pi \[Chi]] (1 + O[1/z])) + (((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[b, 3]])/ (Gamma[Subscript[a, 1]] Gamma[Subscript[b, 1] - Subscript[a, 1]] Gamma[Subscript[b, 2] - Subscript[a, 1]] Gamma[Subscript[b, 3] - Subscript[a, 1]])) (Log[-z] (1 + O[1/z]) - (2 EulerGamma + PolyGamma[Subscript[a, 1]] + PolyGamma[Subscript[b, 1] - Subscript[a, 1]] + PolyGamma[Subscript[b, 2] - Subscript[a, 1]] + PolyGamma[Subscript[b, 3] - Subscript[a, 1]]) (1 + O[1/z])))/ (-z)^Subscript[a, 1] /; (Abs[z] -> Infinity) && \[Chi] == (1/2) (1/2 + 2 Subscript[a, 1] - Subscript[b, 1] - Subscript[b, 2] - Subscript[b, 3])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["b", "1"], "]"]], RowBox[List["Gamma", "[", SubscriptBox["b", "2"], "]"]], RowBox[List["Gamma", "[", SubscriptBox["b", "3"], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["a", "1"], "]"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "\[Chi]"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]], "+", RowBox[List["\[Pi]", " ", "\[Chi]"]]]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["16", " ", SqrtBox[RowBox[List["-", "z"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["16", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["b", "1"], SubscriptBox["b", "2"]]], "+", RowBox[List[SubscriptBox["b", "1"], SubscriptBox["b", "3"]]], "+", RowBox[List[SubscriptBox["b", "2"], SubscriptBox["b", "3"]]], "-", SubsuperscriptBox["a", "1", "2"]]], ")"]]]], "+", RowBox[List["2", RowBox[List["(", RowBox[List[RowBox[List["4", "\[Chi]"]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["6", SubscriptBox["a", "1"]]], "+", SubscriptBox["b", "1"], "+", SubscriptBox["b", "2"], "+", SubscriptBox["b", "3"], "-", "2"]], ")"]]]], "-", "3"]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]], "+", RowBox[List["\[Pi]", " ", "\[Chi]"]]]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["b", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["b", "2"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["b", "3"], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "1"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "1"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "1"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "3"], "-", SubscriptBox["a", "1"]]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["a", "1"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "EulerGamma"]], "+", RowBox[List["PolyGamma", "[", SubscriptBox["a", "1"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "1"]]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "1"]]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["b", "3"], "-", SubscriptBox["a", "1"]]], "]"]]]], ")"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["\[Chi]", "\[Equal]", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["2", SubscriptBox["a", "1"]]], "-", SubscriptBox["b", "1"], "-", SubscriptBox["b", "2"], "-", SubscriptBox["b", "3"]]], ")"]]]]]]]]]]]]

 MathML Form

 2 F 3 ( a 1 , a 1 ; b 1 , b 2 , b 3 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "3"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] Γ ( b 1 ) Γ ( b 2 ) Γ ( b 3 ) π Γ ( a 1 ) 2 ( - z ) χ ( cos ( π χ + 2 - z ) ( 1 + O ( 1 z ) ) + 1 16 - z ( 2 ( 6 a 1 + b 1 + b 2 + b 3 - 2 ) ( 4 χ - 1 ) + 16 ( - a 1 2 + b 1 b 2 + b 1 b 3 + b 2 b 3 ) - 3 ) sin ( π χ + 2 - z ) ( 1 + O ( 1 z ) ) ) + Γ ( b 1 ) Γ ( b 2 ) Γ ( b 3 ) Γ ( a 1 ) Γ ( b 1 - a 1 ) Γ ( b 2 - a 1 ) Γ ( b 3 - a 1 ) ( - z ) - a 1 ( log ( - z ) ( 1 + O ( 1 z ) ) - ( ψ TagBox["\[Psi]", PolyGamma] ( b 1 - a 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( b 2 - a 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( b 3 - a 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( a 1 ) + 2 TagBox["\[DoubledGamma]", Function[EulerGamma]] ) ( 1 + O ( 1 z ) ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) χ 1 2 ( 2 a 1 - b 1 - b 2 - b 3 + 1 2 ) Condition Proportional HypergeometricPFQ Subscript a 1 Subscript a 1 Subscript b 1 Subscript b 2 Subscript b 3 z Gamma Subscript b 1 Gamma Subscript b 2 Gamma Subscript b 3 1 2 Gamma Subscript a 1 2 -1 -1 z χ χ 2 -1 z 1 2 1 O 1 z -1 1 16 -1 z 1 2 -1 2 6 Subscript a 1 Subscript b 1 Subscript b 2 Subscript b 3 -2 4 χ -1 16 -1 Subscript a 1 2 Subscript b 1 Subscript b 2 Subscript b 1 Subscript b 3 Subscript b 2 Subscript b 3 -3 χ 2 -1 z 1 2 1 O 1 z -1 Gamma Subscript b 1 Gamma Subscript b 2 Gamma Subscript b 3 Gamma Subscript a 1 Gamma Subscript b 1 -1 Subscript a 1 Gamma Subscript b 2 -1 Subscript a 1 Gamma Subscript b 3 -1 Subscript a 1 -1 -1 z -1 Subscript a 1 -1 z 1 O 1 z -1 -1 PolyGamma Subscript b 1 -1 Subscript a 1 PolyGamma Subscript b 2 -1 Subscript a 1 PolyGamma Subscript b 3 -1 Subscript a 1 PolyGamma Subscript a 1 2 1 O 1 z -1 Rule z χ 1 2 2 Subscript a 1 -1 Subscript b 1 -1 Subscript b 2 -1 Subscript b 3 1 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", SubscriptBox["a_", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"], ",", SubscriptBox["b_", "3"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["bb", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "2"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "3"], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "\[Chi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]], "+", RowBox[List["\[Pi]", " ", "\[Chi]"]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["16", " ", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["bb", "1"], " ", SubscriptBox["bb", "2"]]], "+", RowBox[List[SubscriptBox["bb", "1"], " ", SubscriptBox["bb", "3"]]], "+", RowBox[List[SubscriptBox["bb", "2"], " ", SubscriptBox["bb", "3"]]], "-", SubsuperscriptBox["aa", "1", "2"]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "\[Chi]"]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", SubscriptBox["aa", "1"]]], "+", SubscriptBox["bb", "1"], "+", SubscriptBox["bb", "2"], "+", SubscriptBox["bb", "3"], "-", "2"]], ")"]]]], "-", "3"]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]], "+", RowBox[List["\[Pi]", " ", "\[Chi]"]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List["16", " ", SqrtBox[RowBox[List["-", "z"]]]]]]]], ")"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["aa", "1"], "]"]], "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["bb", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "2"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "3"], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["aa", "1"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "EulerGamma"]], "+", RowBox[List["PolyGamma", "[", SubscriptBox["aa", "1"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["bb", "1"], "-", SubscriptBox["aa", "1"]]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["bb", "2"], "-", SubscriptBox["aa", "1"]]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["bb", "3"], "-", SubscriptBox["aa", "1"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]]]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["aa", "1"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["bb", "1"], "-", SubscriptBox["aa", "1"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["bb", "2"], "-", SubscriptBox["aa", "1"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["bb", "3"], "-", SubscriptBox["aa", "1"]]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List["\[Chi]", "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["2", " ", SubscriptBox["aa", "1"]]], "-", SubscriptBox["bb", "1"], "-", SubscriptBox["bb", "2"], "-", SubscriptBox["bb", "3"]]], ")"]]]]]]]]]]]]]]

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

 2001-10-29