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/03.16.06.0007.01

 Input Form

 KelvinKer[z] == (1/2) Sum[((-1 + I)^k/(2^((3 k)/2) k!)) (Sum[Binomial[k, 2 j] (I (1 - I^k) (-2 I (-1)^k Pi Floor[Arg[z - x]/(2 Pi)] KelvinBei[-4 j + k, x] + KelvinKei[4 j - k, x]) + (1 + I^k) (-2 I (-1)^k Pi Floor[Arg[z - x]/(2 Pi)] KelvinBer[-4 j + k, x] + KelvinKer[4 j - k, x])), {j, 0, Floor[k/2]}] - Sum[Binomial[k, 1 + 2 j] (I (1 - I^k) (-2 I (-1)^k Pi Floor[Arg[z - x]/(2 Pi)] KelvinBei[-2 - 4 j + k, x] + KelvinKei[2 + 4 j - k, x]) + (1 + I^k) (-2 I (-1)^k Pi Floor[Arg[z - x]/(2 Pi)] KelvinBer[-2 - 4 j + k, x] + KelvinKer[2 + 4 j - k, x])), {j, 0, Floor[(k - 1)/2]}]) (z - x)^k, {k, 0, Infinity}] /; Element[x, Reals] && x < 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKer", "[", "z", "]"]], "\[Equal]", "\n", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[ImaginaryI]"]], ")"]], "k"], " ", SuperscriptBox["2", RowBox[List["-", FractionBox[RowBox[List["3", " ", "k"]], "2"]]]]]], RowBox[List["k", "!"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox["k", "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", RowBox[List["2", " ", "j"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "j"]], "+", "k"]], ",", "x"]], "]"]]]], "+", RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List[RowBox[List["4", " ", "j"]], "-", "k"]], ",", "x"]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "j"]], "+", "k"]], ",", "x"]], "]"]]]], "+", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List[RowBox[List["4", " ", "j"]], "-", "k"]], ",", "x"]], "]"]]]], ")"]]]]]], ")"]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["k", "-", "1"]], "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", RowBox[List["1", "+", RowBox[List["2", " ", "j"]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["4", " ", "j"]], "+", "k"]], ",", "x"]], "]"]]]], "+", RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["4", " ", "j"]], "-", "k"]], ",", "x"]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["4", " ", "j"]], "+", "k"]], ",", "x"]], "]"]]]], "+", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["4", " ", "j"]], "-", "k"]], ",", "x"]], "]"]]]], ")"]]]]]], ")"]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", "<", "0"]]]]]]]]

 MathML Form

 ker ( z ) 1 2 k = 0 ( - 1 + ) k 2 - 3 k 2 k ! ( j = 0 k 2 ( k 2 j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List["2", " ", "j"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( ( 1 - k ) ( kei 4 j - k ( x ) - 2 ( - 1 ) k π arg ( z - x ) 2 π bei k - 4 j ( x ) ) + ( 1 + k ) ( ker 4 j - k ( x ) - 2 ( - 1 ) k π arg ( z - x ) 2 π ber k - 4 j ( x ) ) ) - j = 0 k - 1 2 ( k 2 j + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( ( 1 - k ) ( kei 4 j - k + 2 ( x ) - 2 ( - 1 ) k π arg ( z - x ) 2 π bei - 4 j + k - 2 ( x ) ) + ( 1 + k ) ( ker 4 j - k + 2 ( x ) - 2 ( - 1 ) k π arg ( z - x ) 2 π ber - 4 j + k - 2 ( x ) ) ) ) ( z - x ) k /; x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] x < 0 Condition KelvinKer z 1 2 k 0 -1 k 2 -1 3 k 2 -1 k -1 j 0 k 2 -1 Binomial k 2 j 1 -1 k KelvinKei 4 j -1 k x -1 2 -1 k z -1 x 2 -1 KelvinBei k -1 4 j x 1 k KelvinKer 4 j -1 k x -1 2 -1 k z -1 x 2 -1 KelvinBer k -1 4 j x -1 j 0 k -1 2 -1 Binomial k 2 j 1 1 -1 k KelvinKei 4 j -1 k 2 x -1 2 -1 k z -1 x 2 -1 KelvinBei -4 j k -2 x 1 k KelvinKer 4 j -1 k 2 x -1 2 -1 k z -1 x 2 -1 KelvinBer -4 j k -2 x z -1 x k x x 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKer", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[ImaginaryI]"]], ")"]], "k"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["3", " ", "k"]], ")"]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox["k", "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", RowBox[List["2", " ", "j"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "j"]], "+", "k"]], ",", "x"]], "]"]]]], "+", RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List[RowBox[List["4", " ", "j"]], "-", "k"]], ",", "x"]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "j"]], "+", "k"]], ",", "x"]], "]"]]]], "+", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List[RowBox[List["4", " ", "j"]], "-", "k"]], ",", "x"]], "]"]]]], ")"]]]]]], ")"]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["k", "-", "1"]], "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", RowBox[List["1", "+", RowBox[List["2", " ", "j"]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["4", " ", "j"]], "+", "k"]], ",", "x"]], "]"]]]], "+", RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["4", " ", "j"]], "-", "k"]], ",", "x"]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["4", " ", "j"]], "+", "k"]], ",", "x"]], "]"]]]], "+", RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["4", " ", "j"]], "-", "k"]], ",", "x"]], "]"]]]], ")"]]]]]], ")"]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", "0"]]]]]]]]]]

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

 2007-05-02