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 Hypergeometric0F1

 http://functions.wolfram.com/07.17.06.0013.01

 Input Form

 Hypergeometric0F1[b, z] \[Proportional] (Gamma[b]/Sqrt[Pi]) (-z)^((1 - 2 b)/4) ((1 + ((3 - 2 b) (5 - 2 b) (-1 + 2 b) (1 + 2 b))/ (512 z) + (1/(1572864 z^2)) ((-9 + 2 b) (-7 + 2 b) (-5 + 2 b) (-3 + 2 b) (-1 + 2 b) (1 + 2 b) (3 + 2 b) (5 + 2 b)) + \[Ellipsis]) Cos[(1/4) (-1 + 2 b) Pi - 2 Sqrt[-z]] + (((-3 + 2 b) (-1 + 2 b))/(16 Sqrt[-z])) (1 + ((5 - 2 b) (7 - 2 b) (1 + 2 b) (3 + 2 b))/(1536 z) + (1/(7864320 z^2)) ((-11 + 2 b) (-9 + 2 b) (-7 + 2 b) (-5 + 2 b) (1 + 2 b) (3 + 2 b) (5 + 2 b) (7 + 2 b)) + \[Ellipsis]) Sin[(1/4) (-1 + 2 b) Pi - 2 Sqrt[-z]]) /; (Abs[z] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric0F1", "[", RowBox[List["b", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List["Gamma", "[", "b", "]"]], SqrtBox["\[Pi]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], FractionBox[RowBox[List["1", "-", RowBox[List["2", "b"]]]], "4"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]]]], RowBox[List["512", " ", "z"]]], "+", RowBox[List[FractionBox["1", RowBox[List["1572864", " ", SuperscriptBox["z", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "9"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "b"]]]], ")"]]]], ")"]]]], "+", "\[Ellipsis]"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", "\[Pi]"]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " "]], RowBox[List["16", " ", SqrtBox[RowBox[List["-", "z"]]]]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["7", "-", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "b"]]]], ")"]]]], RowBox[List["1536", " ", "z"]]], "+", RowBox[List[FractionBox["1", RowBox[List["7864320", " ", SuperscriptBox["z", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "11"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "9"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["7", "+", RowBox[List["2", " ", "b"]]]], ")"]]]], ")"]]]], "+", "\[Ellipsis]"]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", "\[Pi]"]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]

 MathML Form

 0 F 1 ( ; b ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["0", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["b", Hypergeometric0F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric0F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric0F1] Γ ( b ) ( - z ) 1 4 ( 1 - 2 b ) π ( ( 1 + ( 3 - 2 b ) ( 5 - 2 b ) ( - 1 + 2 b ) ( 1 + 2 b ) 512 z + 1 1572864 z 2 ( ( - 9 + 2 b ) ( - 7 + 2 b ) ( - 5 + 2 b ) ( - 3 + 2 b ) ( - 1 + 2 b ) ( 1 + 2 b ) ( 3 + 2 b ) ( 5 + 2 b ) ) + ) cos ( ( 2 b - 1 ) π 4 - 2 - z ) + ( 2 b - 3 ) ( 2 b - 1 ) 16 - z ( 1 + ( 5 - 2 b ) ( 7 - 2 b ) ( 1 + 2 b ) ( 3 + 2 b ) 1536 z + 1 7864320 z 2 ( ( - 11 + 2 b ) ( - 9 + 2 b ) ( - 7 + 2 b ) ( - 5 + 2 b ) ( 1 + 2 b ) ( 3 + 2 b ) ( 5 + 2 b ) ( 7 + 2 b ) ) + ) sin ( ( 2 b - 1 ) π 4 - 2 - z ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional Hypergeometric0F1 b z Gamma b -1 z 1 4 1 -1 2 b 1 2 -1 1 3 -1 2 b 5 -1 2 b -1 2 b 1 2 b 512 z -1 1 1572864 z 2 -1 -9 2 b -7 2 b -5 2 b -3 2 b -1 2 b 1 2 b 3 2 b 5 2 b 2 b -1 4 -1 -1 2 -1 z 1 2 2 b -3 2 b -1 16 -1 z 1 2 -1 1 5 -1 2 b 7 -1 2 b 1 2 b 3 2 b 1536 z -1 1 7864320 z 2 -1 -11 2 b -9 2 b -7 2 b -5 2 b 1 2 b 3 2 b 5 2 b 7 2 b 2 b -1 4 -1 -1 2 -1 z 1 2 Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric0F1", "[", RowBox[List["b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "b", "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "b"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]]]], RowBox[List["512", " ", "z"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "9"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "b"]]]], ")"]]]], RowBox[List["1572864", " ", SuperscriptBox["z", "2"]]]], "+", "\[Ellipsis]"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", "\[Pi]"]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "b"]]]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["7", "-", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "b"]]]], ")"]]]], RowBox[List["1536", " ", "z"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "11"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "9"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", RowBox[List["(", RowBox[List["7", "+", RowBox[List["2", " ", "b"]]]], ")"]]]], RowBox[List["7864320", " ", SuperscriptBox["z", "2"]]]], "+", "\[Ellipsis]"]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "b"]]]], ")"]], " ", "\[Pi]"]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], "]"]]]], RowBox[List["16", " ", SqrtBox[RowBox[List["-", "z"]]]]]]]], ")"]]]], SqrtBox["\[Pi]"]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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

 2001-10-29