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StruveH






Mathematica Notation

Traditional Notation









Bessel-Type Functions > StruveH[nu,z] > Differentiation > Symbolic differentiation > With respect to z





http://functions.wolfram.com/03.09.20.0019.01









  


  










Input Form





D[StruveH[\[Nu], z], {z, n}] == (n!/(-(z/2))^n) Sum[((-1)^k/(2^(2 k) k! (n - 2 k)!)) Sum[Binomial[n - k, p] Pochhammer[-(\[Nu]/2), n - k - p] ((z/2)^p StruveH[\[Nu] + p, z] - (1/Pi) (z/2)^(\[Nu] + 2 p - 1) Sum[Gamma[r + 1/2]/Gamma[\[Nu] + p - r + 1/2]/(z/2)^(2 r), {r, 0, p - 1}]), {p, 0, n - k}], {k, 0, Floor[n/2]}] /; Element[n, Integers] && n >= 0










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> n </mi> </msup> <mrow> <msub> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveH </ci> </annotation-xml> </semantics> </mstyle> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mtext> </mtext> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mtext> </mtext> </mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> p </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> p </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[&quot;n&quot;, &quot;-&quot;, &quot;k&quot;]], Identity, Rule[Editable, True]]], List[TagBox[&quot;p&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; 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</mi> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> &#960; </mi> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> p </mi> </mrow> <mo> + </mo> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> r </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> r </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> p </mi> <mo> - </mo> <mi> r </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> r </mi> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <ci> StruveH </ci> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <factorial /> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> p </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <ci> p </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> p </ci> </apply> <apply> <ci> StruveH </ci> <apply> <plus /> <ci> p </ci> <ci> &#957; </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <pi /> <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 /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> p </ci> </apply> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> r </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> r </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </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> <times /> <cn type='integer'> -2 </cn> <ci> r </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["StruveH", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["n", "-", "k"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "k"]], ",", "p"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["n", "-", "k", "-", "p"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "p"], " ", RowBox[List["StruveH", "[", RowBox[List[RowBox[List["\[Nu]", "+", "p"]], ",", "z"]], "]"]]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "p"]], "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], RowBox[List["p", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["r", "+", FractionBox["1", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "r"]]]]], RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "p", "-", "r", "+", FractionBox["1", "2"]]], "]"]]]]]]], "\[Pi]"]]], ")"]]]]]]]], RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["z", "2"]]], ")"]], "n"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










Contributed by





Brychkov Yu.A. (2005)










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





2007-05-02