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; }

 KelvinKer

 http://functions.wolfram.com/03.20.06.0025.01

 Input Form

 KelvinKer[n, z] \[Proportional] ((2^(-2 - n) Pi z^n Sin[(3 n Pi)/4])/n!) (1 - z^4/(32 (1 + n) (2 + n)) + z^8/(6144 (1 + n) (2 + n) (3 + n) (4 + n)) + \[Ellipsis]) + ((2^(-4 - n) Pi z^(2 + n) Cos[(3 n Pi)/4])/ (n + 1)!) (1 - z^4/(96 (2 + n) (3 + n)) + z^8/(30720 (2 + n) (3 + n) (4 + n) (5 + n)) + \[Ellipsis]) + ((1/4) Sum[(((E^((3 I Pi n)/4) + (-1)^k/E^((3 I Pi n)/4)) (n - k - 1)!)/ k!) ((I z^2)/4)^k, {k, 0, n - 1}])/(z/2)^n - 2^(-2 - n) (-1)^n z^n (((2 Cos[(n Pi)/4])/n!) (EulerGamma + 2 Log[z/2] - PolyGamma[1 + n]) + (Sin[(n Pi)/4]/(2 (1 + n)!)) (-1 + EulerGamma + 2 Log[z/2] - PolyGamma[2 + n]) z^2 - (Cos[(n Pi)/4]/(32 (2 + n)!)) (-(3/2) + EulerGamma + 2 Log[z/2] - PolyGamma[3 + n]) z^4 + \[Ellipsis]) /; (z -> 0) && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKer", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "n"]]], " ", "\[Pi]", " ", SuperscriptBox["z", "n"], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", "n", " ", "\[Pi]"]], "4"], "]"]]]], RowBox[List["n", "!"]]], RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["32", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]]]]], "+", FractionBox[SuperscriptBox["z", "8"], RowBox[List["6144", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "n"]], ")"]]]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "4"]], "-", "n"]]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", "+", "n"]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["3", " ", "n", " ", "\[Pi]"]], "4"], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]], RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["96", " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "n"]], ")"]]]]], "+", FractionBox[SuperscriptBox["z", "8"], RowBox[List["30720", " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", "n"]], ")"]]]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[FractionBox["1", "4"], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", "\[ImaginaryI]", " ", "\[Pi]", " ", "n"]], "4"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["3", "\[ImaginaryI]", " ", "\[Pi]", " ", "n"]], "4"]]], " "]]]]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]]]], RowBox[List["k", "!"]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["z", "n"], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "4"], "]"]]]], RowBox[List["n", "!"]]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "n"]], "]"]]]], ")"]]]], "+", " ", RowBox[List[FractionBox[RowBox[List["Sin", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "4"], "]"]], RowBox[List["2", " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], "!"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "EulerGamma", "+", RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", "n"]], "]"]]]], ")"]], SuperscriptBox["z", "2"]]], "-", " ", RowBox[List[FractionBox[RowBox[List["Cos", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "4"], "]"]], RowBox[List["32", " ", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], "!"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "+", "EulerGamma", "+", RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["3", "+", "n"]], "]"]]]], ")"]], SuperscriptBox["z", "4"]]], "+", "\[Ellipsis]"]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 ker n ( z ) 1 4 ( z 2 ) - n k = 0 n - 1 ( 3 π n 4 + ( - 1 ) k - 3 π n 4 ) ( n - k - 1 ) ! k ! ( z 2 4 ) k - 2 - n - 2 ( - 1 ) n z n ( 2 cos ( n π 4 ) n ! ( 2 log ( z 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( n + 1 ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) + sin ( n π 4 ) 2 ( n + 1 ) ! ( 2 log ( z 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( n + 2 ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] - 1 ) z 2 - cos ( n π 4 ) 32 ( n + 2 ) ! ( 2 log ( z 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( n + 3 ) - 3 2 + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) z 4 + ) + 2 - n - 2 π z n sin ( 3 n π 4 ) n ! ( 1 - z 4 32 ( n + 1 ) ( n + 2 ) + z 8 6144 ( n + 1 ) ( n + 2 ) ( n + 3 ) ( n + 4 ) + ) + 2 - n - 4 π z n + 2 cos ( 3 n π 4 ) ( n + 1 ) ! ( 1 - z 4 96 ( n + 2 ) ( n + 3 ) + z 8 30720 ( n + 2 ) ( n + 3 ) ( n + 4 ) ( n + 5 ) + ) /; ( z "\[Rule]" 0 ) n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition Proportional KelvinKer n z 1 4 z 2 -1 -1 n k 0 n -1 3 n 4 -1 -1 k -1 3 n 4 -1 n -1 k -1 k -1 z 2 4 -1 k -1 2 -1 n -2 -1 n z n 2 n 4 -1 n -1 2 z 2 -1 -1 PolyGamma n 1 n 4 -1 2 n 1 -1 2 z 2 -1 -1 PolyGamma n 2 -1 z 2 -1 n 4 -1 32 n 2 -1 2 z 2 -1 -1 PolyGamma n 3 -1 3 2 z 4 2 -1 n -2 z n 3 n 4 -1 n -1 1 -1 z 4 32 n 1 n 2 -1 z 8 6144 n 1 n 2 n 3 n 4 -1 2 -1 n -4 z n 2 3 n 4 -1 n 1 -1 1 -1 z 4 96 n 2 n 3 -1 z 8 30720 n 2 n 3 n 4 n 5 -1 Rule z 0 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKer", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "n"]]], " ", "\[Pi]", " ", SuperscriptBox["z", "n"], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", "n", " ", "\[Pi]"]], "4"], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["32", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]]]]], "+", FractionBox[SuperscriptBox["z", "8"], RowBox[List["6144", " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "n"]], ")"]]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["n", "!"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "4"]], "-", "n"]]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", "+", "n"]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["3", " ", "n", " ", "\[Pi]"]], "4"], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["96", " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "n"]], ")"]]]]], "+", FractionBox[SuperscriptBox["z", "8"], RowBox[List["30720", " ", RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", "n"]], ")"]]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]], "+", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "n"]], "4"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "n"]], ")"]]]]]]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["z", "n"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "4"], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "n"]], "]"]]]], ")"]]]], RowBox[List["n", "!"]]], "+", FractionBox[RowBox[List[RowBox[List["Sin", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "4"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "EulerGamma", "+", RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", "n"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "2"]]], RowBox[List["2", " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], "!"]]]]], "-", FractionBox[RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "4"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "+", "EulerGamma", "+", RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["3", "+", "n"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "4"]]], RowBox[List["32", " ", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], "!"]]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02