Struve function L: Identities (formula 03.10.17.0010)

StruveL

$Mathematica Notation$

Bessel-Type Functions

StruveL[nu,z]

Identities

Recurrence identities

Distant neighbors

Decreasing

http://functions.wolfram.com/03.10.17.0010.01

Input Form

StruveL[\[Nu], z] == -((2^(1 - \[Nu]) z^(-7 + \[Nu]) (z^6 + z^4 (1 - 2 \[Nu]) - 4 z^2 (24 - 82 \[Nu] + 83 \[Nu]^2 - 32 \[Nu]^3 + 4 \[Nu]^4) - 8 (90 - 441 \[Nu] + 812 \[Nu]^2 - 735 \[Nu]^3 + 350 \[Nu]^4 - 84 \[Nu]^5 + 8 \[Nu]^6)))/(Sqrt[Pi] Gamma[1/2 + \[Nu]])) - (4 (-2 + \[Nu]) (6 + z^2 - 8 \[Nu] + 2 \[Nu]^2) StruveL[-5 + \[Nu], z])/ z^3 + (1 + (4 (-2 + \[Nu]) (-1 + \[Nu]))/z^2 + (8 (-4 + \[Nu]) (-2 + \[Nu]) (6 + z^2 - 8 \[Nu] + 2 \[Nu]^2))/z^4) StruveL[-4 + \[Nu], z]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["StruveL", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "7"]], "+", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", "6"], "+", RowBox[List[SuperscriptBox["z", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "-", RowBox[List["4", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["24", "-", RowBox[List["82", " ", "\[Nu]"]], "+", RowBox[List["83", " ", SuperscriptBox["\[Nu]", "2"]]], "-", RowBox[List["32", " ", SuperscriptBox["\[Nu]", "3"]]], "+", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "4"]]]]], ")"]]]], "-", RowBox[List["8", " ", RowBox[List["(", RowBox[List["90", "-", RowBox[List["441", " ", "\[Nu]"]], "+", RowBox[List["812", " ", SuperscriptBox["\[Nu]", "2"]]], "-", RowBox[List["735", " ", SuperscriptBox["\[Nu]", "3"]]], "+", RowBox[List["350", " ", SuperscriptBox["\[Nu]", "4"]]], "-", RowBox[List["84", " ", SuperscriptBox["\[Nu]", "5"]]], "+", RowBox[List["8", " ", SuperscriptBox["\[Nu]", "6"]]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]], ")"]]]]]], "-", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["6", "+", SuperscriptBox["z", "2"], "-", RowBox[List["8", " ", "\[Nu]"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]], " ", RowBox[List["StruveL", "[", RowBox[List[RowBox[List[RowBox[List["-", "5"]], "+", "\[Nu]"]], ",", "z"]], "]"]]]], SuperscriptBox["z", "3"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]]]], SuperscriptBox["z", "2"]], "+", FractionBox[RowBox[List["8", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "4"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["6", "+", SuperscriptBox["z", "2"], "-", RowBox[List["8", " ", "\[Nu]"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]]]], SuperscriptBox["z", "4"]]]], ")"]], " ", RowBox[List["StruveL", "[", RowBox[List[RowBox[List[RowBox[List["-", "4"]], "+", "\[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> L </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveL </ci> </annotation-xml> </semantics> </mstyle> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <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> <mi> ν </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <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> </mrow> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> L </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveL </ci> </annotation-xml> </semantics> </mstyle> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 4 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <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> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> L </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveL </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> 3 </mn> </msup> </mfrac> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 7 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 6 </mn> </msup> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 32 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 83 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 82 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 24 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 6 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 84 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 350 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 735 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 812 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 441 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 90 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> StruveL </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -4 </cn> </apply> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </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'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <ci> ν </ci> </apply> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> StruveL </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -4 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </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'> -1 </cn> <apply> <times /> <cn type='integer'> 8 </cn> <ci> ν </ci> </apply> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <ci> StruveL </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'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <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'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -7 </cn> </apply> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 83 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 82 </cn> <ci> ν </ci> </apply> </apply> <cn type='integer'> 24 </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'> 8 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 84 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 350 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 735 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 812 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 441 </cn> <ci> ν </ci> </apply> </apply> <cn type='integer'> 90 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["StruveL", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "7"]], "+", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", "6"], "+", RowBox[List[SuperscriptBox["z", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], "-", RowBox[List["4", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List["24", "-", RowBox[List["82", " ", "\[Nu]"]], "+", RowBox[List["83", " ", SuperscriptBox["\[Nu]", "2"]]], "-", RowBox[List["32", " ", SuperscriptBox["\[Nu]", "3"]]], "+", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "4"]]]]], ")"]]]], "-", RowBox[List["8", " ", RowBox[List["(", RowBox[List["90", "-", RowBox[List["441", " ", "\[Nu]"]], "+", RowBox[List["812", " ", SuperscriptBox["\[Nu]", "2"]]], "-", RowBox[List["735", " ", SuperscriptBox["\[Nu]", "3"]]], "+", RowBox[List["350", " ", SuperscriptBox["\[Nu]", "4"]]], "-", RowBox[List["84", " ", SuperscriptBox["\[Nu]", "5"]]], "+", RowBox[List["8", " ", SuperscriptBox["\[Nu]", "6"]]]]], ")"]]]]]], ")"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], "]"]]]]]]], "-", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["6", "+", SuperscriptBox["z", "2"], "-", RowBox[List["8", " ", "\[Nu]"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]], " ", RowBox[List["StruveL", "[", RowBox[List[RowBox[List[RowBox[List["-", "5"]], "+", "\[Nu]"]], ",", "z"]], "]"]]]], SuperscriptBox["z", "3"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]]]], SuperscriptBox["z", "2"]], "+", FractionBox[RowBox[List["8", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "4"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["6", "+", SuperscriptBox["z", "2"], "-", RowBox[List["8", " ", "\[Nu]"]], "+", RowBox[List["2", " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]]]], SuperscriptBox["z", "4"]]]], ")"]], " ", RowBox[List["StruveL", "[", RowBox[List[RowBox[List[RowBox[List["-", "4"]], "+", "\[Nu]"]], ",", "z"]], "]"]]]]]]]]]]

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

2002-12-18