html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 KelvinKei

 http://functions.wolfram.com/03.19.20.0004.01

 Input Form

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

 Standard Form

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 MathML Form

 kei TagBox["kei", BesselJ] n + 1 2 ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( z ) 1 8 π ( π bei n + 1 2 ( z ) - 3 ( - 1 ) n π ber - n - 1 2 ( z ) - 4 ( log ( z ) - log ( - 1 4 z ) ) ( ( - 1 ) n bei - n - 1 2 ( z ) - ber n + 1 2 ( z ) ) ) - ( - 1 ) 3 / 8 2 - n - 5 2 n π 4 π z - n - 1 2 n ! - - 1 4 z 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 ) ! ( ( - 1 ) n 2 ( - 1 + ) ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) + 2 - 1 4 z ( Chi ( ( 1 + ) 2 z ) - Shi ( ( 1 + ) 2 z ) ) ) + 2 ( - 1 ) k π n 2 + 2 z ( Ci ( ( 1 + ) 2 z ) + 2 ( - 1 ) 3 / 4 z ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) + Si ( ( 1 + ) 2 z ) ) ) k 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 ) ! ( ( - 1 ) 3 / 4 ( - 1 ) n ( - 1 ) 3 / 4 z ( - ( 1 + ) 2 z Chi ( ( 1 + ) 2 z ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 3 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) + 2 - 1 4 z Shi ( 2 - 1 4 z ) ) + ( - 1 ) k n π 2 sin ( - 1 4 z ) ( Ci ( ( 1 + ) 2 z ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 3 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) + Si ( ( 1 + ) 2 z ) ) - ( - 1 ) k n π 2 cos ( - 1 4 z ) ( Ci ( ( 1 + ) 2 z ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 3 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) + Si ( ( 1 + ) 2 z ) ) ) k z 2 k /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition 1 0 Subscript BesselJ kei n 1 2 z 1 8 KelvinBei n 1 2 z -1 3 -1 n KelvinBer -1 n -1 1 2 z -1 4 z -1 -1 1 4 z -1 n KelvinBei -1 n -1 1 2 z -1 KelvinBer n 1 2 z -1 -1 3 8 2 -1 n -1 5 2 n 4 -1 1 2 z -1 n -1 1 2 n -1 -1 -1 1 4 z k 0 n 2 -1 2 2 k Binomial n 2 k 2 n -1 2 k -1 n 2 1 2 -1 PolyGamma k 1 2 -1 PolyGamma k -1 n 1 2 2 -1 1 4 z CoshIntegral 1 2 1 2 z -1 SinhIntegral 1 2 1 2 z 2 -1 k n 2 -1 2 1 2 z CosIntegral 1 2 1 2 z 2 -1 3 4 z PolyGamma k 1 2 -1 PolyGamma k -1 n 1 2 SinIntegral 1 2 1 2 z k z 2 k -1 -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 -1 3 4 -1 n -1 3 4 z -1 1 2 1 2 z CoshIntegral 1 2 1 2 z PolyGamma k 3 2 -1 PolyGamma k -1 n 1 2 2 -1 1 4 z SinhIntegral 2 -1 1 4 z -1 k n 2 -1 -1 1 4 z CosIntegral 1 2 1 2 z PolyGamma k 3 2 -1 PolyGamma k -1 n 1 2 SinIntegral 1 2 1 2 z -1 -1 k n 2 -1 -1 1 4 z CosIntegral 1 2 1 2 z -1 PolyGamma k 3 2 PolyGamma k -1 n 1 2 SinIntegral 1 2 1 2 z k z 2 k n [/itex]

 Rule Form

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

 2007-05-02