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 KelvinKer

 http://functions.wolfram.com/03.20.20.0005.01

 Input Form

 Derivative[1, 0][KelvinKer][-n - 1/2, z] == (Pi/8) (-3 (-1)^n Pi KelvinBei[1/2 + n, z] + Pi KelvinBer[-(1/2) - n, z] - 4 (Log[z] - Log[(-1)^(1/4) z]) (KelvinBei[-(1/2) - n, z] - (-1)^n KelvinBer[1/2 + n, z])) + (((-1)^(3/8) 2^(-(3/2) - n) E^((1/4) I n Pi) Sqrt[Pi] z^(-(1/2) - n))/n!) Sum[2^(2 k) Binomial[n, 2 k] (2 n - 2 k)! I^k (((CoshIntegral[2 (-1)^(1/4) z] - PolyGamma[1/2 + k] + PolyGamma[1/2 + k - n]) Sinh[(-1)^(1/4) z] - Cosh[(-1)^(1/4) z] SinhIntegral[2 (-1)^(1/4) z])/(-1)^4^(-1) + (Cosh[(-1)^(1/4) z] (CoshIntegral[2 (-1)^(1/4) z] + PolyGamma[1/2 + k] - PolyGamma[1/2 + k - n]) - Sinh[(-1)^(1/4) z] SinhIntegral[2 (-1)^(1/4) z])/(-1)^4^(-1) - (-1)^k E^((3 I n Pi)/2) ((CosIntegral[2 (-1)^(1/4) z] - PolyGamma[1/2 + k] + PolyGamma[1/2 + k - n]) Sin[(-1)^(1/4) z] - Cos[(-1)^(1/4) z] SinIntegral[2 (-1)^(1/4) z]) - I (-1)^k E^((3 I n Pi)/2) (Cos[(-1)^(1/4) z] (CosIntegral[2 (-1)^(1/4) z] + PolyGamma[1/2 + k] - PolyGamma[1/2 + k - n]) + Sin[(-1)^(1/4) z] SinIntegral[2 (-1)^(1/4) z])) z^(2 k), {k, 0, Floor[n/2]}] + (((-1)^(5/8) 2^(-(1/2) - n) E^((1/4) I n Pi) Sqrt[Pi] z^(1/2 - n))/n!) Sum[2^(2 k) Binomial[n, 2 k + 1] (2 n - 2 k - 1)! I^k ((-(-1)^(-4^(-1))) ((CoshIntegral[2 (-1)^(1/4) z] + PolyGamma[3/2 + k] - PolyGamma[1/2 + k - n]) Sinh[(-1)^(1/4) z] - Cosh[(-1)^(1/4) z] SinhIntegral[2 (-1)^(1/4) z]) - (Cosh[(-1)^(1/4) z] (CoshIntegral[2 (-1)^(1/4) z] - PolyGamma[3/2 + k] + PolyGamma[1/2 + k - n]) - Sinh[(-1)^(1/4) z] SinhIntegral[2 (-1)^(1/4) z])/(-1)^4^(-1) - I (-1)^k E^((3 I n Pi)/2) ((CosIntegral[2 (-1)^(1/4) z] + PolyGamma[3/2 + k] - PolyGamma[1/2 + k - n]) Sin[(-1)^(1/4) z] - Cos[(-1)^(1/4) z] SinIntegral[2 (-1)^(1/4) z]) + (-1)^k E^((3 I n Pi)/2) (Cos[(-1)^(1/4) z] (CosIntegral[2 (-1)^(1/4) z] - PolyGamma[3/2 + k] + PolyGamma[1/2 + k - n]) + Sin[(-1)^(1/4) z] SinIntegral[2 (-1)^(1/4) z])) z^(2 k), {k, 0, Floor[(n - 1)/2]}] /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 ker TagBox["ker", BesselJ] - n - 1 2 ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( z ) 1 8 π ( π ber - n - 1 2 ( z ) - 4 ( log ( z ) - log ( - 1 4 z ) ) ( bei - n - 1 2 ( z ) - ( - 1 ) n ber n + 1 2 ( z ) ) - 3 ( - 1 ) n π bei n + 1 2 ( z ) ) + ( - 1 ) 3 / 8 2 - n - 3 2 n π 4 π z - n - 1 2 n ! k = 0 n 2 2 2 k ( n 2 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List["2", " ", "k"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( 2 n - 2 k ) ! k ( 1 - 1 4 ( ( Chi ( 2 - 1 4 z ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) sinh ( - 1 4 z ) - cosh ( - 1 4 z ) Shi ( 2 - 1 4 z ) ) + 1 - 1 4 ( cosh ( - 1 4 z ) ( Chi ( 2 - 1 4 z ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) - sinh ( - 1 4 z ) Shi ( 2 - 1 4 z ) ) - ( - 1 ) k 3 n π 2 ( cos ( - 1 4 z ) ( Ci ( 2 - 1 4 z ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) + sin ( - 1 4 z ) Si ( 2 - 1 4 z ) ) - ( - 1 ) k 3 n π 2 ( ( Ci ( 2 - 1 4 z ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) sin ( - 1 4 z ) - cos ( - 1 4 z ) Si ( 2 - 1 4 z ) ) ) z 2 k + ( - 1 ) 5 / 8 2 - n - 1 2 n π 4 π z 1 2 - n n ! k = 0 n - 1 2 2 2 k ( n 2 k + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( - 2 k + 2 n - 1 ) ! k ( - 1 - 1 4 ( ( Chi ( 2 - 1 4 z ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 3 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) sinh ( - 1 4 z ) - cosh ( - 1 4 z ) Shi ( 2 - 1 4 z ) ) - 1 - 1 4 ( cosh ( - 1 4 z ) ( Chi ( 2 - 1 4 z ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 3 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) - sinh ( - 1 4 z ) Shi ( 2 - 1 4 z ) ) + ( - 1 ) k 3 n π 2 ( cos ( - 1 4 z ) ( Ci ( 2 - 1 4 z ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 3 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) + sin ( - 1 4 z ) Si ( 2 - 1 4 z ) ) - ( - 1 ) k 3 n π 2 ( ( Ci ( 2 - 1 4 z ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 3 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) sin ( - 1 4 z ) - cos ( - 1 4 z ) Si ( 2 - 1 4 z ) ) ) z 2 k /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition 1 0 Subscript BesselJ ker -1 n -1 1 2 z 1 8 KelvinBer -1 n -1 1 2 z -1 4 z -1 -1 1 4 z KelvinBei -1 n -1 1 2 z -1 -1 n KelvinBer n 1 2 z -1 3 -1 n KelvinBei n 1 2 z -1 3 8 2 -1 n -1 3 2 n 4 -1 1 2 z -1 n -1 1 2 n -1 k 0 n 2 -1 2 2 k Binomial n 2 k 2 n -1 2 k k 1 -1 1 4 -1 CoshIntegral 2 -1 1 4 z -1 PolyGamma k 1 2 PolyGamma k -1 n 1 2 -1 1 4 z -1 -1 1 4 z SinhIntegral 2 -1 1 4 z 1 -1 1 4 -1 -1 1 4 z CoshIntegral 2 -1 1 4 z PolyGamma k 1 2 -1 PolyGamma k -1 n 1 2 -1 -1 1 4 z SinhIntegral 2 -1 1 4 z -1 -1 k 3 n 2 -1 -1 1 4 z CosIntegral 2 -1 1 4 z PolyGamma k 1 2 -1 PolyGamma k -1 n 1 2 -1 1 4 z SinIntegral 2 -1 1 4 z -1 -1 k 3 n 2 -1 CosIntegral 2 -1 1 4 z -1 PolyGamma k 1 2 PolyGamma k -1 n 1 2 -1 1 4 z -1 -1 1 4 z SinIntegral 2 -1 1 4 z z 2 k -1 5 8 2 -1 n -1 1 2 n 4 -1 1 2 z 1 2 -1 n n -1 k 0 n -1 2 -1 2 2 k Binomial n 2 k 1 -2 k 2 n -1 k -1 1 -1 1 4 -1 CoshIntegral 2 -1 1 4 z PolyGamma k 3 2 -1 PolyGamma k -1 n 1 2 -1 1 4 z -1 -1 1 4 z SinhIntegral 2 -1 1 4 z -1 1 -1 1 4 -1 -1 1 4 z CoshIntegral 2 -1 1 4 z -1 PolyGamma k 3 2 PolyGamma k -1 n 1 2 -1 -1 1 4 z SinhIntegral 2 -1 1 4 z -1 k 3 n 2 -1 -1 1 4 z CosIntegral 2 -1 1 4 z -1 PolyGamma k 3 2 PolyGamma k -1 n 1 2 -1 1 4 z SinIntegral 2 -1 1 4 z -1 -1 k 3 n 2 -1 CosIntegral 2 -1 1 4 z PolyGamma k 3 2 -1 PolyGamma k -1 n 1 2 -1 1 4 z -1 -1 1 4 z SinIntegral 2 -1 1 4 z z 2 k n [/itex]

 Rule Form

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

 2007-05-02