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/03.20.06.0026.01

 Input Form

 KelvinKer[n, z] == (Pi/4) Sum[(Sin[(Pi (2 k + 3 n))/4]/(k! (k + n)!)) (z/2)^(2 k + n), {k, 0, Infinity}] + ((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 Sum[((E^(-((I Pi n)/4)) + (-1)^k E^((I Pi n)/4))/ (k! (k + n)!)) (2 Log[z/2] - PolyGamma[1 + k] - PolyGamma[1 + k + n]) ((I z^2)/4)^k, {k, 0, Infinity}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKer", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["\[Pi]", "4"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["3", "n"]]]], ")"]]]], "4"], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "n"]]]]]]]]], "+", 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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "n"]], "4"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "n"]], "4"]]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k", "+", "n"]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[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 + π 4 k = 0 sin ( 1 4 π ( 2 k + 3 n ) ) k ! ( k + n ) ! ( z 2 ) 2 k + n - 2 - n - 2 ( - 1 ) n z n k = 0 ( - π n 4 + ( - 1 ) k π n 4 ) ( 2 log ( z 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + n + 1 ) ) k ! ( k + n ) ! ( z 2 4 ) k 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 4 -1 k 0 1 4 2 k 3 n k k n -1 z 2 -1 2 k n -1 2 -1 n -2 -1 n z n k 0 -1 n 4 -1 -1 k n 4 -1 2 z 2 -1 -1 PolyGamma k 1 -1 PolyGamma k n 1 k k n -1 z 2 4 -1 k [/itex]

 Rule Form

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

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

 2007-05-02