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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





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MathML Form







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) </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;, 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<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> 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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> 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Contributed by





Brychkov Yu.A. (2005)










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





2007-05-02