Struve function H: Series representations (formula 03.09.06.0010)

StruveH

$Mathematica Notation$

Bessel-Type Functions

StruveH[nu,z]

Series representations

Asymptotic series expansions

Expansions for any z in trigonometric form

Using trigonometric functions with branch cut-containing arguments

http://functions.wolfram.com/03.09.06.0010.01

Input Form

StruveH[\[Nu], z] \[Proportional] (Sqrt[2/Pi] z^(\[Nu] + 1) (Sin[Sqrt[z^2] - ((2 \[Nu] + 1)/4) Pi] HypergeometricPFQ[{(1 - 2 \[Nu])/4, (3 - 2 \[Nu])/4, (1 + 2 \[Nu])/4, (3 + 2 \[Nu])/4}, {1/2}, -(1/z^2)] + ((4 \[Nu]^2 - 1)/(8 Sqrt[z^2])) Cos[Sqrt[z^2] - ((2 \[Nu] + 1)/4) Pi] HypergeometricPFQ[ {(3 - 2 \[Nu])/4, (5 - 2 \[Nu])/4, (3 + 2 \[Nu])/4, (5 + 2 \[Nu])/4}, {3/2}, -(1/z^2)]))/(z^2)^((3 + 2 \[Nu])/4) + ((2^(1 - \[Nu]) z^(\[Nu] - 1))/(Sqrt[Pi] Gamma[1/2 + \[Nu]])) HypergeometricPFQ[{1/2, 1/2 - \[Nu], 1}, {}, -(4/z^2)] /; (Abs[z] -> Infinity)

Standard Form

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MathML Form

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<ci> ν </ci> </apply> </apply> <cn type='integer'> 1 </cn> </list> <list /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </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["StruveH", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SqrtBox[FractionBox["2", "\[Pi]"]], " ", SuperscriptBox["z", RowBox[List["\[Nu]", "+", "1"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], "-", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]], " ", "\[Pi]"]]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]], "-", "1"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], "-", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]], " ", "\[Pi]"]]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "]"]]]], RowBox[List["8", " ", SqrtBox[SuperscriptBox["z", "2"]]]]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["\[Nu]", "-", "1"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", "1"]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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

2001-10-29