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 HypergeometricPFQ

 http://functions.wolfram.com/07.22.06.0012.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1]}, {Subscript[b, 1], Subscript[b, 2]}, z] \[Proportional] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] (((-z)^((1/2) (1/2 + Subscript[a, 1] - Subscript[b, 1] - Subscript[b, 2]))/ (Sqrt[Pi] Gamma[Subscript[a, 1]])) (Cos[2 Sqrt[-z] + (Pi/2) (1/2 + Subscript[a, 1] - Subscript[b, 1] - Subscript[b, 2])] (1 + O[1/z]) + (1/(16 Sqrt[-z])) (12 Subscript[a, 1]^2 - 4 Subscript[b, 1]^2 + 8 Subscript[b, 2] - 4 Subscript[b, 2]^2 + 8 Subscript[b, 1] (Subscript[b, 2] + 1) - 8 Subscript[a, 1] (Subscript[b, 1] + Subscript[b, 2] + 1) - 3) Sin[2 Sqrt[-z] + (Pi/2) (1/2 + Subscript[a, 1] - Subscript[b, 1] - Subscript[b, 2])] (1 + O[1/z])) + (1/((-z)^Subscript[a, 1] (Gamma[Subscript[b, 1] - Subscript[a, 1]] Gamma[Subscript[b, 2] - Subscript[a, 1]]))) (1 + O[1/z])) /; (Abs[z] -> Infinity)

 Standard Form

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 MathML Form

 1 F 2 ( a 1 ; b 1 , b 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[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]]]], 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 ) ( ( - z ) - a 1 Γ ( b 1 - a 1 ) Γ ( b 2 - a 1 ) ( 1 + O ( 1 z ) ) + 1 π Γ ( a 1 ) ( - z ) 1 2 ( a 1 - b 1 - b 2 + 1 2 ) ( cos ( π 2 ( a 1 - b 1 - b 2 + 1 2 ) + 2 - z ) ( 1 + O ( 1 z ) ) + 1 16 - z ( ( 12 a 1 2 - 8 ( b 1 + b 2 + 1 ) a 1 - 4 b 1 2 - 4 b 2 2 + 8 ( b 2 + 1 ) b 1 + 8 b 2 - 3 ) sin ( π 2 ( a 1 - b 1 - b 2 + 1 2 ) + 2 - z ) ( 1 + O ( 1 z ) ) ) ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional HypergeometricPFQ Subscript a 1 Subscript b 1 Subscript b 2 z Gamma Subscript b 1 Gamma Subscript b 2 -1 z -1 Subscript a 1 Gamma Subscript b 1 -1 Subscript a 1 Gamma Subscript b 2 -1 Subscript a 1 -1 1 O 1 z -1 1 1 2 Gamma Subscript a 1 -1 -1 z 1 2 Subscript a 1 -1 Subscript b 1 -1 Subscript b 2 1 2 2 -1 Subscript a 1 -1 Subscript b 1 -1 Subscript b 2 1 2 2 -1 z 1 2 1 O 1 z -1 1 16 -1 z 1 2 -1 12 Subscript a 1 2 -1 8 Subscript b 1 Subscript b 2 1 Subscript a 1 -1 4 Subscript b 1 2 -1 4 Subscript b 2 2 8 Subscript b 2 1 Subscript b 1 8 Subscript b 2 -3 2 -1 Subscript a 1 -1 Subscript b 1 -1 Subscript b 2 1 2 2 -1 z 1 2 1 O 1 z -1 Rule z [/itex]

 Rule Form

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

 2001-10-29