Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

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
 











StruveH






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/03.09.20.0016.01









  


  










Input Form





Derivative[1, 0][StruveH][n + 1/2, z] == (-2 SinIntegral[z] + SinIntegral[2 z]) BesselJ[n + 1/2, z] + (-1)^n (CosIntegral[2 z] - 2 CosIntegral[z]) BesselJ[-n - 1/2, z] + (1/(n! Sqrt[Pi])) (z/2)^(n - 1/2) Log[z/2] HypergeometricPFQ[{-n, 1/2, 1}, {}, -(4/z^2)] + (1/(2 Pi)) (z/2)^(-(1/2) - n) Gamma[n + 1/2] (3 EulerGamma + Log[4] + PolyGamma[1/2 - n]) - (n!/2) (2/z)^n Sum[(1/((n - k) k!)) (-(z/2))^k BesselJ[-k - 1/2, z], {k, 0, n - 1}] - (n!/(2 Sqrt[Pi])) (z/2)^(-(1/2) - n) Sum[Pochhammer[1/2, k]/(k! (n - k)), {k, 0, n - 1}] - (1/Sqrt[Pi]) (z/2)^(n - 1/2) Sum[((2/z)^(2 k) Pochhammer[1/2, k] PolyGamma[n - k + 1])/(n - k)!, {k, 0, n - 1}] - (1/2) n! Sqrt[Pi] (z/2)^(1/2 - n) Sum[(1/(k! (n - k))) (z/2)^k Sum[(1/p!) (z/2)^p ((-1)^(p + 1) BesselJ[k + 1/2, z] (2 BesselJ[1/2 - p, z] - 2^(1/2 + p) BesselJ[1/2 - p, 2 z]) - (-1)^k BesselJ[-k - 1/2, z] (2 BesselJ[p - 1/2, z] - 2^(1/2 + p) BesselJ[p - 1/2, 2 z])), {p, 0, n - k - 1}], {k, 0, n - 1}] /; Element[n, Integers] && n >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0"]], "]"]], "[", "StruveH", "]"]], "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["SinIntegral", "[", "z", "]"]]]], "+", RowBox[List["SinIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List[RowBox[List["CosIntegral", "[", RowBox[List["2", " ", "z"]], "]"]], "-", RowBox[List["2", " ", RowBox[List["CosIntegral", "[", "z", "]"]]]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["n", "!"]], " ", SqrtBox["\[Pi]"]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["n", "-", FractionBox["1", "2"]]]], " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", FractionBox["1", "2"], ",", "1"]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", " ", "\[Pi]"]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List["n", "+", FractionBox["1", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "EulerGamma"]], "+", RowBox[List["Log", "[", "4", "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "n"]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["n", "!"]]]], "2"], SuperscriptBox[RowBox[List["(", FractionBox["2", "z"], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["z", "2"]]], ")"]], "k"], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]]]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["n", "!"]], " "]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]]]]]]]]], "-", RowBox[List[FractionBox["1", SqrtBox["\[Pi]"]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["n", "-", FractionBox["1", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "z"], ")"]], RowBox[List["2", " ", "k"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["n", "-", "k", "+", "1"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]]]]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["n", "!"]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[FractionBox["1", "2"], "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["n", "-", "k", "-", "1"]]], RowBox[List[FractionBox["1", RowBox[List["p", "!"]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "p"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "+", "1"]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["k", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "p"]], ",", "z"]], "]"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "+", "p"]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "p"]], ",", RowBox[List["2", " ", "z"]]]], "]"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "-", FractionBox["1", "2"]]], ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["p", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "+", "p"]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["p", "-", FractionBox["1", "2"]]], ",", RowBox[List["2", " ", "z"]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mstyle fontweight='bold' fontstyle='normal'> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> StruveH </ci> </annotation-xml> </semantics> </mstyle> <mrow> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;,&quot;, &quot;0&quot;]], &quot;)&quot;]], Derivative] </annotation> </semantics> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <mtext> </mtext> <mn> 1 </mn> </mrow> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 3 </mn> </msub> <msub> <mi> F </mi> <mn> 0 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mo> &#8202; </mo> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mn> 4 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, &quot;3&quot;], SubscriptBox[&quot;F&quot;, &quot;0&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;-&quot;, &quot;n&quot;]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;1&quot;, HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[&quot;\[Null]&quot;, InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;4&quot;, 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> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <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> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> p </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mi> p </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> p </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mrow> <mi> k </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> p </mi> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> p </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> p </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mrow> <mi> p </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> p </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mrow> <mi> p </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Ci </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Ci </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msub> <mi> J </mi> <mrow> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Si </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> Si </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mtext> </mtext> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </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> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 0 </cn> </list> <apply> <ci> Subscript </ci> <ci> StruveH </ci> <apply> <plus /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </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> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='integer'> 1 </cn> </list> <list /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </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> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </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> PolyGamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <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 /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </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 /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </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> <ci> k </ci> </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> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <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> <power /> <apply> <factorial /> <ci> p </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> BesselJ </ci> <apply> <plus /> <ci> k </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> BesselJ </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> p </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> BesselJ </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> p </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <ci> BesselJ </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> BesselJ </ci> <apply> <plus /> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> p </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> BesselJ </ci> <apply> <plus /> <ci> p </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <ln /> <cn type='integer'> 4 </cn> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <eulergamma /> </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'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <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> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </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> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <ci> BesselJ </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> CosIntegral </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> CosIntegral </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> BesselJ </ci> <apply> <plus /> <ci> n </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> SinIntegral </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> SinIntegral </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> k </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 /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <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> k </ci> </apply> <apply> <ci> BesselJ </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> z </ci> </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[SuperscriptBox["StruveH", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["n_", "+", FractionBox["1", "2"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["SinIntegral", "[", "z", "]"]]]], "+", RowBox[List["SinIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List[RowBox[List["CosIntegral", "[", RowBox[List["2", " ", "z"]], "]"]], "-", RowBox[List["2", " ", RowBox[List["CosIntegral", "[", "z", "]"]]]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["n", "-", FractionBox["1", "2"]]]], " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", FractionBox["1", "2"], ",", "1"]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["-", FractionBox["4", SuperscriptBox["z", "2"]]]]]], "]"]]]], RowBox[List[RowBox[List["n", "!"]], " ", SqrtBox["\[Pi]"]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List["n", "+", FractionBox["1", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "EulerGamma"]], "+", RowBox[List["Log", "[", "4", "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "n"]], "]"]]]], ")"]]]], RowBox[List["2", " ", "\[Pi]"]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["n", "!"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["2", "z"], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["z", "2"]]], ")"]], "k"], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], "-", FractionBox[RowBox[List[RowBox[List["n", "!"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]]]]]]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["n", "-", FractionBox["1", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "z"], ")"]], RowBox[List["2", " ", "k"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["n", "-", "k", "+", "1"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]]]]]]], SqrtBox["\[Pi]"]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["n", "!"]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[FractionBox["1", "2"], "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["n", "-", "k", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "p"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "+", "1"]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["k", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "p"]], ",", "z"]], "]"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "+", "p"]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "p"]], ",", RowBox[List["2", " ", "z"]]]], "]"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "-", FractionBox["1", "2"]]], ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["p", "-", FractionBox["1", "2"]]], ",", "z"]], "]"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "+", "p"]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["p", "-", FractionBox["1", "2"]]], ",", RowBox[List["2", " ", "z"]]]], "]"]]]]]], ")"]]]]]], ")"]]]], RowBox[List["p", "!"]]]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]]]]]]]]]]], "/;", 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





© 1998-2014 Wolfram Research, Inc.