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StruveL






Mathematica Notation

Traditional Notation









Bessel-Type Functions > StruveL[nu,z] > Identities > Recurrence identities > Distant neighbors > Decreasing





http://functions.wolfram.com/03.10.17.0019.01









  


  










Input Form





StruveL[\[Nu], z] == Subscript[\[ScriptCapitalC], n][\[Nu], z] StruveL[\[Nu] - n, z] + Subscript[\[ScriptCapitalC], n - 1][\[Nu], z] StruveL[\[Nu] - n - 1, z] - (1/Sqrt[Pi]) (z/2)^(\[Nu] - 1) Sum[(Pochhammer[1 - \[Nu], j]/(Gamma[-j + \[Nu] + 1/2] (z^2/4)^j)) HypergeometricPFQ[{(1 - j)/2, -(j/2)}, {1 - \[Nu], -j, -j + \[Nu]}, z^2], {j, 0, n - 1}] /; Subscript[\[ScriptCapitalC], n][\[Nu], z] == (2^n Pochhammer[1 - \[Nu], n] HypergeometricPFQ[{(1 - n)/2, -(n/2)}, {1 - \[Nu], -n, -n + \[Nu]}, z^2])/z^n && Element[n, Integers] && n > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["StruveL", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], RowBox[List["StruveL", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], RowBox[List["StruveL", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n", "-", "1"]], ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox["1", SqrtBox["\[Pi]"]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["\[Nu]", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "j"]], "]"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "j"]], "+", "\[Nu]", "+", FractionBox["1", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], "j"]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "j"]], "2"], ",", RowBox[List["-", FractionBox["j", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["-", "j"]], ",", RowBox[List[RowBox[List["-", "j"]], "+", "\[Nu]"]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", "n"], " ", SuperscriptBox["z", RowBox[List["-", "n"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "n"]], "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", RowBox[List["-", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["-", "n"]], "+", "\[Nu]"]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]]]], "\[And]", RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]










MathML Form







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</mo> <mrow> <mrow> <msub> <mi> &#119966; </mi> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <msup> <mn> 2 </mn> <mi> n </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;\[Nu]&quot;]], &quot;)&quot;]], &quot;n&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mrow> <mi> &#957; </mi> <mo> - </mo> <mi> n </mi> </mrow> </mrow> <mo> ; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, &quot;2&quot;], SubscriptBox[&quot;F&quot;, &quot;3&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;n&quot;]], &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;n&quot;, &quot;2&quot;]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;\[Nu]&quot;]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;n&quot;]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;\[Nu]&quot;, &quot;-&quot;, &quot;n&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[SuperscriptBox[&quot;z&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> StruveL </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <ci> n </ci> </apply> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <ci> StruveL </ci> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <ci> StruveL </ci> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <ci> j </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> j </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> j </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </list> <list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; 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</ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </list> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </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[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["StruveL", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["StruveL", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n", "-", "1"]], ",", "z"]], "]"]]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["\[Nu]", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "j"]], "2"], ",", RowBox[List["-", FractionBox["j", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["-", "j"]], ",", RowBox[List[RowBox[List["-", "j"]], "+", "\[Nu]"]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "j"]], "+", "\[Nu]", "+", FractionBox["1", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["z", "2"], "4"], ")"]], "j"]]]]]]]], SqrtBox["\[Pi]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", "n"], " ", SuperscriptBox["z", RowBox[List["-", "n"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", "n"]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", RowBox[List["-", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["-", "n"]], "+", "\[Nu]"]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










Contributed by





Brychkov Yu.A. (2005)










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





2007-05-02