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StruveL






Mathematica Notation

Traditional Notation









Bessel-Type Functions > StruveL[nu,z] > Series representations > Generalized power series > Expansions at generic point z==z0 > For the function itself





http://functions.wolfram.com/03.10.06.0019.01









  


  










Input Form





StruveL[\[Nu], z] == Sqrt[Pi] Gamma[2 + \[Nu]] (Subscript[z, 0]/4)^(\[Nu] + 1) (1/Subscript[z, 0])^ (\[Nu] Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) Subscript[z, 0]^(\[Nu] Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) Sum[(2^k/(Subscript[z, 0]^k k!)) HypergeometricPFQRegularized[ {1, 1 + \[Nu]/2, (3 + \[Nu])/2}, {3/2, 1 + (\[Nu] - k)/2, (3 - k + \[Nu])/2, 3/2 + \[Nu]}, Subscript[z, 0]^2/4] (z - Subscript[z, 0])^k, {k, 0, Infinity}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["StruveL", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SqrtBox["\[Pi]"], RowBox[List["Gamma", "[", RowBox[List["2", "+", "\[Nu]"]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox[SubscriptBox["z", "0"], "4"], ")"]], RowBox[List["\[Nu]", "+", "1"]]], SuperscriptBox[RowBox[List["(", FractionBox["1", SubscriptBox["z", "0"]], ")"]], RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SubsuperscriptBox["z", "0", RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[SuperscriptBox["2", "k"], " ", SubsuperscriptBox["z", "0", RowBox[List["-", "k"]]]]]]], RowBox[List["k", "!"]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["1", "+", FractionBox[RowBox[List["\[Nu]", "-", "k"]], "2"]]], ",", FractionBox[RowBox[List["3", "-", "k", "+", "\[Nu]"]], "2"], ",", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]]]], "}"]], ",", FractionBox[SubsuperscriptBox["z", "0", "2"], "4"]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]










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> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </msubsup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mi> k </mi> </msup> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msubsup> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 3 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 4 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mi> k </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mfrac> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;3&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;4&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;1&quot;, HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;\[Nu]&quot;, &quot;2&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;\[Nu]&quot;, &quot;+&quot;, &quot;3&quot;]], &quot;2&quot;], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;3&quot;, &quot;2&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[RowBox[List[&quot;\[Nu]&quot;, &quot;-&quot;, &quot;k&quot;]], &quot;2&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot; &quot;, RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[&quot;-&quot;, &quot;k&quot;]], &quot;+&quot;, &quot;\[Nu]&quot;, &quot;+&quot;, &quot;3&quot;]], &quot;)&quot;]]]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;\[Nu]&quot;, &quot;+&quot;, FractionBox[&quot;3&quot;, &quot;2&quot;]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[FractionBox[SubsuperscriptBox[&quot;z&quot;, &quot;0&quot;, &quot;2&quot;], &quot;4&quot;], HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> StruveL </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <cn type='integer'> 1 </cn> <apply> <plus /> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> &#957; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </list> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <ci> k </ci> </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[SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", "\[Nu]"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SubscriptBox["zz", "0"], "4"], ")"]], RowBox[List["\[Nu]", "+", "1"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", SubscriptBox["zz", "0"]], ")"]], RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SubsuperscriptBox["zz", "0", RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", "k"], " ", SubsuperscriptBox["zz", "0", RowBox[List["-", "k"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["3", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["1", "+", FractionBox[RowBox[List["\[Nu]", "-", "k"]], "2"]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["3", "-", "k", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["3", "2"], "+", "\[Nu]"]]]], "}"]], ",", FractionBox[SubsuperscriptBox["zz", "0", "2"], "4"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]]]]]]










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





2007-05-02





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