Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











StruveL




Mathematica Notation

Traditional Notation









Bessel-Type Functions > StruveL[nu,z] > Integral representations > Contour integral representations





http://functions.wolfram.com/03.10.07.0004.01









  


  








Input Form





StruveL[\[Nu], z] == (-Pi) Csc[(Pi \[Nu])/2] z^(\[Nu] - 1) (z^2)^((1 - \[Nu])/2) (1/(2 Pi I)) ContourIntegral[ (Gamma[s + (\[Nu] + 1)/2] Gamma[(1 - \[Nu])/2 - s])/ (Gamma[s + 1/2] Gamma[1/2 - s] Gamma[1 + \[Nu]/2 - s] Gamma[1 - \[Nu]/2 - s])/(z^2/4)^s, \[ScriptCapitalL]["s"]]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["StruveL", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", RowBox[List["Csc", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], RowBox[List[" ", "2"]]], "]"]], " ", SuperscriptBox["z", RowBox[List["\[Nu]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]], FractionBox["1", RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List[SubscriptBox["\[Integral]", "\[ScriptCapitalL]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "-", "s"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["1", "2"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox["\[Nu]", "2"], "-", "s"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["\[Nu]", "2"], "-", "s"]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], RowBox[List["-", "s"]]], RowBox[List["\[DifferentialD]", "s"]]]]]]]]]]]]








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> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> &#8290; </mo> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mo> &#8747; </mo> <mi> &#8466; </mi> </msub> <mrow> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> s </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> s </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> InterpretationBox </ci> <ms> L </ms> <ci> StruveL </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> <apply> <ci> Rule </ci> <ci> Selectable </ci> <false /> </apply> </apply> <ms> &#957; </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> &#960; </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> csc </ms> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#960; </ms> <ms> &#957; </ms> </list> </apply> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <apply> <ci> RowBox </ci> <list> <ms> &#957; </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> &#960; </ms> <ms> &#8520; </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> &#8747; </ms> </apply> <ms> &#8466; </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#957; </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> 2 </ms> </apply> <ms> + </ms> <ms> s </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> &#957; </ms> </list> </apply> <ms> 2 </ms> </apply> <ms> - </ms> <ms> s </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> s </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> <ms> - </ms> <ms> s </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> &#957; </ms> <ms> 2 </ms> </apply> <ms> - </ms> <ms> s </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> &#957; </ms> <ms> 2 </ms> </apply> <ms> - </ms> <ms> s </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <ms> 4 </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> s </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> &#8518; </ms> <ms> s </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["StruveL", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["Csc", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], " ", SuperscriptBox["z", RowBox[List["\[Nu]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]], " ", RowBox[List["ContourIntegral", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "-", "s"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], RowBox[List["-", "s"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["s", "+", FractionBox["1", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox["\[Nu]", "2"], "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["\[Nu]", "2"], "-", "s"]], "]"]]]]], ",", RowBox[List["\[ScriptCapitalL]", "[", "\"s\"", "]"]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]]]]]]]]


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





2001-10-29