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/03.19.06.0028.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2007-05-02