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 Hypergeometric0F1Regularized

 http://functions.wolfram.com/07.18.03.0002.01

 Input Form

 Hypergeometric0F1Regularized[b, z] == (-(1/Sqrt[Pi])) E^((1/2) Pi I (3/2 - b)) z^((1 - 2 b)/4) (Sinh[((Pi I)/2) (3/2 - b) - 2 Sqrt[z]] Sum[(Abs[b - 1] + 2 k - 1/2)!/(2^(4 k) (2 k)! (Abs[b - 1] - 2 k - 1/2)! z^k), {k, 0, Floor[(1/4) (2 Abs[b - 1] - 1)]}] + (1/Sqrt[z]) Cosh[((Pi I)/2) (3/2 - b) - 2 Sqrt[z]] Sum[(Abs[b - 1] + 2 k + 1/2)!/(2^(4 k + 2) (2 k + 1)! (Abs[b - 1] - 2 k - 3/2)! z^k), {k, 0, Floor[(1/4) (2 Abs[b - 1] - 3)]}]) /; Element[b - 1/2, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List["b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", SqrtBox["\[Pi]"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[FractionBox["3", "2"], "-", "b"]], ")"]]]]], SuperscriptBox["z", FractionBox[RowBox[List["1", "-", RowBox[List["2", "b"]]]], "4"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sinh", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"], RowBox[List["(", RowBox[List[FractionBox["3", "2"], "-", "b"]], ")"]]]], "-", RowBox[List["2", SqrtBox["z"]]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]]]], "-", "1"]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "+", RowBox[List["2", "k"]], "-", FractionBox["1", "2"]]], ")"]], "!"]], RowBox[List[SuperscriptBox["2", RowBox[List["4", "k"]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "-", RowBox[List["2", "k"]], "-", FractionBox["1", "2"]]], ")"]], "!"]], " ", SuperscriptBox["z", "k"]]]]]]]], "+", RowBox[List[FractionBox["1", SqrtBox["z"]], RowBox[List["Cosh", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"], RowBox[List["(", RowBox[List[FractionBox["3", "2"], "-", "b"]], ")"]]]], "-", RowBox[List["2", SqrtBox["z"]]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]]]], "-", "3"]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "+", RowBox[List["2", "k"]], "+", FractionBox["1", "2"]]], ")"]], "!"]], " "]], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["4", "k"]], "+", "2"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "-", RowBox[List["2", "k"]], "-", FractionBox["3", "2"]]], ")"]], "!"]], SuperscriptBox["z", "k"]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["b", "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]]]]]]

 MathML Form

 0 F ~ 1 ( ; b ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["0", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["b", Hypergeometric0F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1Regularized, Rule[Editable, False]], ";", TagBox["z", Hypergeometric0F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric0F1Regularized] - 1 π exp ( π 2 ( 3 2 - b ) ) z 1 - 2 b 4 ( sinh ( π 2 ( 3 2 - b ) - 2 z ) k = 0 1 4 ( 2 "\[LeftBracketingBar]" b - 1 "\[RightBracketingBar]" - 1 ) ( 2 k + "\[LeftBracketingBar]" b - 1 "\[RightBracketingBar]" - 1 2 ) ! 2 4 k ( 2 k ) ! ( "\[LeftBracketingBar]" b - 1 "\[RightBracketingBar]" - 2 k - 1 2 ) ! z k + 1 z cosh ( π 2 ( 3 2 - b ) - 2 z ) k = 0 1 4 ( 2 "\[LeftBracketingBar]" b - 1 "\[RightBracketingBar]" - 3 ) ( 2 k + "\[LeftBracketingBar]" b - 1 "\[RightBracketingBar]" + 1 2 ) ! 2 4 k + 2 ( 2 k + 1 ) ! ( "\[LeftBracketingBar]" b - 1 "\[RightBracketingBar]" - 2 k - 3 2 ) ! z k ) /; b - 1 2 TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Hypergeometric0F1Regularized b z -1 1 1 2 -1 2 -1 3 2 -1 b z 1 -1 2 b 4 -1 2 -1 3 2 -1 b -1 2 z 1 2 k 0 1 4 2 b -1 -1 2 k b -1 -1 1 2 2 4 k 2 k b -1 -1 2 k -1 1 2 z k -1 1 z 1 2 -1 2 -1 3 2 -1 b -1 2 z 1 2 k 0 1 4 2 b -1 -3 2 k b -1 1 2 2 4 k 2 2 k 1 b -1 -1 2 k -1 3 2 z k -1 b -1 1 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List["b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[FractionBox["3", "2"], "-", "b"]], ")"]]]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "b"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sinh", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox["3", "2"], "-", "b"]], ")"]]]], "-", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]]]], "-", "1"]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "+", RowBox[List["2", " ", "k"]], "-", FractionBox["1", "2"]]], ")"]], "!"]], RowBox[List[SuperscriptBox["2", RowBox[List["4", " ", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "-", RowBox[List["2", " ", "k"]], "-", FractionBox["1", "2"]]], ")"]], "!"]], " ", SuperscriptBox["z", "k"]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["Cosh", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox["3", "2"], "-", "b"]], ")"]]]], "-", RowBox[List["2", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]]]], "-", "3"]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "+", RowBox[List["2", " ", "k"]], "+", FractionBox["1", "2"]]], ")"]], "!"]], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["4", " ", "k"]], "+", "2"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", RowBox[List["b", "-", "1"]], "]"]], "-", RowBox[List["2", " ", "k"]], "-", FractionBox["3", "2"]]], ")"]], "!"]], " ", SuperscriptBox["z", "k"]]]]]]]], SqrtBox["z"]]]], ")"]]]], SqrtBox["\[Pi]"]]]], "/;", RowBox[List[RowBox[List["b", "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]]]]]]]]

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

 2001-10-29