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 KelvinKer

 http://functions.wolfram.com/03.20.20.0012.01

 Input Form

 Derivative[m, 0][KelvinKer][\[Nu], z] == (-(1/2)) Pi (Sum[(1/k!) (z/2)^(2 k) D[((z/2)^\[Nu] Sin[(1/4) Pi (2 k + 3 \[Nu])])/Gamma[k + \[Nu] + 1], {\[Nu], m}], {k, 0, Infinity}] - Sum[Binomial[m, j] (Pi^(m - j) (-I)^(m - j + 1) Sum[(1/2^i) (((-1)^i i!) StirlingS2[m - j, i] ((I Cot[(Pi \[Nu])/2] + 1)^i (I Cot[(Pi \[Nu])/2] - 1) - 2^(m - j) (I Cot[Pi \[Nu]] + 1)^i (I Cot[Pi \[Nu]] - 1))), {i, 0, m - j}]) Sum[(1/k!) (z/2)^(2 k) D[Cos[(1/4) Pi (2 k - 3 \[Nu])]/((z/2)^\[Nu] Gamma[k - \[Nu] + 1]), {\[Nu], j}], {k, 0, Infinity}], {j, 0, m}] + Sum[Binomial[m, j] Pi^(m - j) ((-KroneckerDelta[m - j]) I + (-I)^(m - j + 1) 2^(m - j) (I Cot[Pi \[Nu]] - 1) Sum[((-1)^i i! StirlingS2[m - j, i] (I Cot[Pi \[Nu]] + 1)^i)/2^i, {i, 0, m - j}]) Sum[(1/k!) (z/2)^(2 k) D[((z/2)^\[Nu] Cos[(1/4) Pi (2 k + 3 \[Nu])])/Gamma[k + \[Nu] + 1], {\[Nu], j}], {k, 0, Infinity}], {j, 0, m}]) /; !Element[\[Nu], Integers]

 Standard Form

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 MathML Form

 ker TagBox["ker", BesselI] ν ( m , 0 ) TagBox[RowBox[List["(", RowBox[List["m", ",", "0"]], ")"]], Derivative] ( z ) - π 2 ( k = 0 1 k ! ( z 2 ) 2 k m ( z 2 ) ν sin ( 1 4 π ( 2 k + 3 ν ) ) Γ ( k + ν + 1 ) ν m - j = 0 m ( m j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["j", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( π m - j ( - ) - j + m + 1 i = 0 m - j ( - 1 ) i i ! 𝒮 TagBox["\[ScriptCapitalS]", StirlingS2] m - j ( i ) 2 i ( ( cot ( π ν 2 ) + 1 ) i ( cot ( π ν 2 ) - 1 ) - 2 m - j ( cot ( π ν ) + 1 ) i ( cot ( π ν ) - 1 ) ) ) k = 0 1 k ! ( z 2 ) 2 k j ( z 2 ) - ν cos ( 1 4 π ( 2 k - 3 ν ) ) Γ ( k - ν + 1 ) ν j + j = 0 m ( m j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["j", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] π m - j ( ( - ) - j + m + 1 2 m - j ( cot ( π ν ) - 1 ) i = 0 m - j ( - 1 ) i i ! 𝒮 TagBox["\[ScriptCapitalS]", StirlingS2] m - j ( i ) ( cot ( π ν ) + 1 ) i 2 i - δ KroneckerDelta m - j ) k = 0 1 k ! ( z 2 ) 2 k j ( z 2 ) ν cos ( 1 4 π ( 2 k + 3 ν ) ) Γ ( k + ν + 1 ) ν j ) /; ν TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition m 0 Subscript BesselI ker ν z -1 2 -1 k 0 1 k -1 z 2 -1 2 k ν m z 2 -1 ν 1 4 2 k 3 ν Gamma k ν 1 -1 -1 j 0 m Binomial m j m -1 j -1 -1 j m 1 i 0 m -1 j -1 i i StirlingS2 m -1 j i 2 i -1 ν 2 -1 1 i ν 2 -1 -1 -1 2 m -1 j ν 1 i ν -1 k 0 1 k -1 z 2 -1 2 k ν j z 2 -1 -1 ν 1 4 2 k -1 3 ν Gamma k -1 ν 1 -1 j 0 m Binomial m j m -1 j -1 -1 j m 1 2 m -1 j ν -1 i 0 m -1 j -1 i i StirlingS2 m -1 j i ν 1 i 2 i -1 -1 KroneckerDelta m -1 j k 0 1 k -1 z 2 -1 2 k ν j z 2 -1 ν 1 4 2 k 3 ν Gamma k ν 1 -1 ν [/itex]

 Rule Form

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

 2007-05-02