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BesselI






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/03.02.20.0019.01









  


  










Input Form





Derivative[1, 0][BesselI][n + 1/2, z] == (((-1)^n 2 (2 z)^(-(1/2) - n))/(n! Sqrt[Pi])) Sum[2^(2 k) Binomial[n, 2 k] (2 n - 2 k)! ((-PolyGamma[k + 1/2] + PolyGamma[k - n + 1/2]) Sinh[z] + Sinh[z] CoshIntegral[2 z] - Cosh[z] SinhIntegral[2 z]) z^(2 k), {k, 0, Floor[n/2]}] + (((-1)^n 2 (2 z)^(1/2 - n))/(n! Sqrt[Pi])) Sum[2^(2 k) Binomial[n, 2 k + 1] (2 n - 2 k - 1)! ((PolyGamma[k + 3/2] - PolyGamma[k - n + 1/2]) Cosh[z] - Cosh[z] CoshIntegral[2 z] + Sinh[z] SinhIntegral[2 z]) z^(2 k), {k, 0, Floor[(n - 1)/2]}] /; Element[n, Integers] && n >= 0










Standard Form





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MathML Form







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<apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <sinh /> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <ci> CoshIntegral </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <sinh /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <cosh /> <ci> z </ci> </apply> <apply> <ci> SinhIntegral </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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Contributed by





Brychkov Yu.A. (2005)










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





2007-05-02