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 KelvinKer

 http://functions.wolfram.com/03.20.06.0028.01

 Input Form

 KelvinKer[n, z] == (1/8) ((-I^(n + 1)) Pi BesselI[n, (-1)^(1/4) z] + (-1)^n Pi I BesselJ[n, (-1)^(1/4) z] + 4 (-I)^n BesselK[n, (-1)^(1/4) z] - 2 (-1)^n Pi BesselY[n, (-1)^(1/4) z] - 4 (I^n BesselI[n, (-1)^(1/4) z] + (-1)^n BesselJ[n, (-1)^(1/4) z]) (Log[z] - Log[(-1)^(1/4) z]) + E^((3 I n Pi)/4) n! Sum[(((-1)^(k/4) 2^(1 - k + n) z^(k - n))/ ((k - n) k!)) ((-1)^(k + n/2) BesselI[k, (-1)^(1/4) z] + BesselJ[k, (-1)^(1/4) z]), {k, 0, n - 1}] - ((2^(1 - n) (-1)^(n/4) z^n)/n!) Sum[(1/j) ((-1)^n HypergeometricPFQ[{j}, {1 + j, 1 + n}, -((I z^2)/4)] + I^n HypergeometricPFQ[{j}, {1 + j, 1 + n}, (I z^2)/4]), {j, 1, n}] + Sum[((2^(1 - 2 k + n) I^k z^(2 k - n) (n - k - 1)!)/k!) (I^n (-1)^(n - k) + (-1)^n), {k, 0, n - 1}]/(-1)^(n/4)) /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 ker n ( z ) 1 8 ( - n + 1 π I n ( - 1 4 z ) + ( - 1 ) n π J n ( - 1 4 z ) + 4 ( - ) n K n ( - 1 4 z ) - 2 ( - 1 ) n π Y n ( - 1 4 z ) - 4 ( n I n ( - 1 4 z ) + ( - 1 ) n J n ( - 1 4 z ) ) ( log ( z ) - log ( - 1 4 z ) ) + 3 n π 4 n ! k = 0 n - 1 ( - 1 ) k / 4 2 - k + n + 1 z k - n ( k - n ) k ! ( ( - 1 ) k + n 2 I k ( - 1 4 z ) + J k ( - 1 4 z ) ) - 2 1 - n ( - 1 ) n / 4 z n n ! j = 1 n 1 j ( ( - 1 ) n 1 F 2 ( j ; j + 1 , n + 1 ; - z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["j", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["j", "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] + n 1 F 2 ( j ; j + 1 , n + 1 ; z 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["j", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["j", "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["n", "+", "1"]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] ) + ( - 1 ) - n 4 k = 0 n - 1 ( 2 - 2 k + n + 1 k z 2 k - n ( n - k - 1 ) ! ) ( n ( - 1 ) n - k + ( - 1 ) n ) k ! ) /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition KelvinKer n z 1 8 -1 n 1 BesselI n -1 1 4 z -1 n BesselJ n -1 1 4 z 4 -1 n BesselK n -1 1 4 z -1 2 -1 n BesselY n -1 1 4 z -1 4 n BesselI n -1 1 4 z -1 n BesselJ n -1 1 4 z z -1 -1 1 4 z 3 n 4 -1 n k 0 n -1 -1 k 4 -1 2 -1 k n 1 z k -1 n k -1 n k -1 -1 k n 2 -1 BesselI k -1 1 4 z BesselJ k -1 1 4 z -1 2 1 -1 n -1 n 4 -1 z n n -1 j 1 n 1 j -1 -1 n HypergeometricPFQ j j 1 n 1 -1 z 2 4 -1 n HypergeometricPFQ j j 1 n 1 z 2 4 -1 -1 -1 n 4 -1 k 0 n -1 2 -2 k n 1 k z 2 k -1 n n -1 k -1 n -1 n -1 k -1 n k -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKer", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ImaginaryI]", RowBox[List["n", "+", "1"]]]]], " ", "\[Pi]", " ", RowBox[List["BesselI", "[", RowBox[List["n", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["BesselJ", "[", RowBox[List["n", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], "+", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "\[ImaginaryI]"]], ")"]], "n"], " ", RowBox[List["BesselK", "[", RowBox[List["n", ",", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02