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variants of this functions
Hypergeometric0F1






Mathematica Notation

Traditional Notation









Hypergeometric Functions > Hypergeometric0F1[b,z] > Series representations > Asymptotic series expansions > Expansions for any z in trigonometric form





http://functions.wolfram.com/07.17.06.0014.01









  


  










Input Form





Hypergeometric0F1[b, z] \[Proportional] (Gamma[b]/Sqrt[Pi]) (-z)^((1 - 2 b)/4) (Cos[((2 b - 1)/4) Pi - 2 Sqrt[-z]] HypergeometricPFQ[{(3 - 2 b)/4, (5 - 2 b)/4, (2 b - 1)/4, (2 b + 1)/4}, {1/2}, 1/(4 z)] + (((2 b - 1) (2 b - 3))/(16 Sqrt[-z])) Sin[((2 b - 1)/4) Pi - 2 Sqrt[-z]] HypergeometricPFQ[ {(5 - 2 b)/4, (7 - 2 b)/4, (1 + 2 b)/4, (3 + 2 b)/4}, {3/2}, 1/(4 z)]) /; (Abs[z] -> Infinity)










Standard Form





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MathML Form







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</list> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>










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["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", "1"]], ")"]], " ", "\[Pi]"]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "b"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "b"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", "1"]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", "1"]], ")"]]]]]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", FractionBox["1", RowBox[List["4", " ", "z"]]]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", "3"]], ")"]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", "1"]], ")"]], " ", "\[Pi]"]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "b"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["7", "-", RowBox[List["2", " ", "b"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "b"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", FractionBox["1", RowBox[List["4", " ", "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





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