Struve function H: Identities (formula 03.09.17.0007)

StruveH

$Mathematica Notation$

Bessel-Type Functions

StruveH[nu,z]

Identities

Recurrence identities

Distant neighbors

Increasing

http://functions.wolfram.com/03.09.17.0007.01

Input Form

StruveH[\[Nu], z] == (2^(-5 - \[Nu]) z^(1 + \[Nu]) (6555 - 259 z^2 + z^4 + 9490 \[Nu] - 160 z^2 \[Nu] + 4840 \[Nu]^2 - 24 z^2 \[Nu]^2 + 1040 \[Nu]^3 + 80 \[Nu]^4))/(Sqrt[Pi] Gamma[13/2 + \[Nu]]) + (1/z^5) (2 (3 + \[Nu]) (3 z^4 - 16 z^2 (8 + 6 \[Nu] + \[Nu]^2) + 16 (40 + 78 \[Nu] + 49 \[Nu]^2 + 12 \[Nu]^3 + \[Nu]^4)) StruveH[5 + \[Nu], z]) - (1 - (4 (1 + \[Nu]) (2 + \[Nu]))/z^2 + (8 (2 + \[Nu]) (4 + \[Nu]) (6 - z^2 + 8 \[Nu] + 2 \[Nu]^2))/z^4) StruveH[6 + \[Nu], z]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["StruveH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "5"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["1", "+", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List["6555", "-", RowBox[List["259", " ", SuperscriptBox["z", "2"]]], "+", SuperscriptBox["z", "4"], "+", RowBox[List["9490", " ", "\[Nu]"]], "-", RowBox[List["160", " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]], "+", RowBox[List["4840", " ", SuperscriptBox["\[Nu]", "2"]]], "-", RowBox[List["24", " ", SuperscriptBox["z", "2"], " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["1040", " ", SuperscriptBox["\[Nu]", "3"]]], "+", RowBox[List["80", " ", SuperscriptBox["\[Nu]", "4"]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["13", "2"], "+", "\[Nu]"]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", SuperscriptBox["z", "5"]], RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["16", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["8", "+", RowBox[List["6", " ", "\[Nu]"]], "+", SuperscriptBox["\[Nu]", "2"]]], ")"]]]], "+", RowBox[List["16", " ", RowBox[List["(", RowBox[List["40", "+", RowBox[List["78", " ", "\[Nu]"]], "+", RowBox[List["49", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["12", " ", SuperscriptBox["\[Nu]", "3"]]], "+", SuperscriptBox["\[Nu]", "4"]]], ")"]]]]]], ")"]], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List["5", "+", "\[Nu]"]], ",", "z"]], "]"]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]]]], SuperscriptBox["z", "2"]], "+", FractionBox[RowBox[List["8", " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["6", "-", SuperscriptBox["z", "2"], "+", RowBox[List["8", " ", "\[Nu]"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]]]], SuperscriptBox["z", "4"]]]], ")"]], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List["6", "+", "\[Nu]"]], ",", "z"]], "]"]]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveH </ci> </annotation-xml> </semantics> </mstyle> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 16 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 16 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> ν </mi> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 49 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 78 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 40 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveH </ci> </annotation-xml> </semantics> </mstyle> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 5 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mfrac> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveH </ci> </annotation-xml> </semantics> </mstyle> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 6 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 5 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> - </mo> <mrow> <mn> 24 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 160 </mn> <mo> ⁢ </mo> <mi> ν </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 259 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 80 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1040 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4840 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 9490 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 6555 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 13 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> StruveH </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <plus /> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <ci> ν </ci> </apply> <cn type='integer'> 8 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <plus /> <apply> <power /> <ci> ν </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 49 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 78 </cn> <ci> ν </ci> </apply> <cn type='integer'> 40 </cn> </apply> </apply> </apply> <apply> <ci> StruveH </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 5 </cn> </apply> <ci> z </ci> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <ci> ν </ci> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> StruveH </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 6 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -5 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 160 </cn> <ci> ν </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 259 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 80 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1040 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4840 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9490 </cn> <ci> ν </ci> </apply> <cn type='integer'> 6555 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 13 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </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[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "5"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["1", "+", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List["6555", "-", RowBox[List["259", " ", SuperscriptBox["z", "2"]]], "+", SuperscriptBox["z", "4"], "+", RowBox[List["9490", " ", "\[Nu]"]], "-", RowBox[List["160", " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]], "+", RowBox[List["4840", " ", SuperscriptBox["\[Nu]", "2"]]], "-", RowBox[List["24", " ", SuperscriptBox["z", "2"], " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["1040", " ", SuperscriptBox["\[Nu]", "3"]]], "+", RowBox[List["80", " ", SuperscriptBox["\[Nu]", "4"]]]]], ")"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["13", "2"], "+", "\[Nu]"]], "]"]]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["16", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["8", "+", RowBox[List["6", " ", "\[Nu]"]], "+", SuperscriptBox["\[Nu]", "2"]]], ")"]]]], "+", RowBox[List["16", " ", RowBox[List["(", RowBox[List["40", "+", RowBox[List["78", " ", "\[Nu]"]], "+", RowBox[List["49", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["12", " ", SuperscriptBox["\[Nu]", "3"]]], "+", SuperscriptBox["\[Nu]", "4"]]], ")"]]]]]], ")"]], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List["5", "+", "\[Nu]"]], ",", "z"]], "]"]]]], SuperscriptBox["z", "5"]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]]]], SuperscriptBox["z", "2"]], "+", FractionBox[RowBox[List["8", " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["6", "-", SuperscriptBox["z", "2"], "+", RowBox[List["8", " ", "\[Nu]"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]]]], SuperscriptBox["z", "4"]]]], ")"]], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List["6", "+", "\[Nu]"]], ",", "z"]], "]"]]]]]]]]]]

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

2002-12-18