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variants of this functions
Hypergeometric0F1Regularized




Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric0F1Regularized[b,z] > Specific values > Specialized values > For fixed z and symbolic parameter





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





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MathML Form







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<times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










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