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 KelvinKei

 http://functions.wolfram.com/03.19.06.0027.01

 Input Form

 KelvinKei[\[Nu], z] == (-(Pi/4)) Sum[(Cos[(Pi (2 (k + \[Nu]) + Abs[\[Nu]]))/4]/ (k! (k + Abs[\[Nu]])!)) (z/2)^(2 k + Abs[\[Nu]]), {k, 0, Infinity}] + ((I/4) Sum[(((E^((I Pi (2 \[Nu] + Abs[\[Nu]]))/4) - (-1)^k/E^((I Pi (2 \[Nu] + Abs[\[Nu]]))/4)) (Abs[\[Nu]] - k - 1)!)/ k!) ((I z^2)/4)^k, {k, 0, Abs[\[Nu]] - 1}])/(z/2)^Abs[\[Nu]] + (I/4) ((I z)/2)^Abs[\[Nu]] E^((I Pi \[Nu])/2) Sum[((E^(-((I Pi Abs[\[Nu]])/4)) - (-1)^k E^((I Pi Abs[\[Nu]])/4))/ (k! (k + Abs[\[Nu]])!)) (2 Log[z/2] - PolyGamma[1 + k] - PolyGamma[1 + k + Abs[\[Nu]]]) ((I z^2)/4)^k, {k, 0, Infinity}] /; Element[\[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "4"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", "\[Nu]"]], ")"]]]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], "4"], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]]]]]]]], "+", RowBox[List[FractionBox["\[ImaginaryI]", "4"], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], "4"]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], "4"]]], " "]]]]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", "1"]], ")"]], "!"]]]], RowBox[List["k", "!"]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]]]]]], "+", RowBox[List[FractionBox["\[ImaginaryI]", "4"], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"], ")"]], RowBox[List["Abs", "[", "\[Nu]", "]"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "4"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "4"]]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]]], " ", 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", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]

 MathML Form

 kei ν ( z ) 4 ( z 2 ) - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" k = 0 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 ( 1 4 π ( 2 ν + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) - ( - 1 ) k - 1 4 ( π ( 2 ν + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ) ) ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - k - 1 ) ! k ! ( z 2 4 ) k - π 4 k = 0 cos ( 1 4 π ( 2 ( k + ν ) + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ) k ! ( k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ! ( z 2 ) 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 4 ( z 2 ) "\[LeftBracketingBar]" ν "\[RightBracketingBar]" π ν 2 k = 0 ( - 1 4 ( π "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) - ( - 1 ) k 1 4 π "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ( 2 log ( z 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 ) ) k ! ( k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ! ( z 2 4 ) k /; ν TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition KelvinKei ν z 4 -1 z 2 -1 -1 ν k 0 ν -1 1 4 2 ν ν -1 -1 k -1 1 4 2 ν ν ν -1 k -1 k -1 z 2 4 -1 k -1 4 -1 k 0 1 4 2 k ν ν k k ν -1 z 2 -1 2 k ν 4 -1 z 2 -1 ν ν 2 -1 k 0 -1 1 4 ν -1 -1 k 1 4 ν 2 z 2 -1 -1 PolyGamma k 1 -1 PolyGamma k ν 1 k k ν -1 z 2 4 -1 k ν [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["-", "\[Pi]"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["k", "+", "\[Nu]"]], ")"]]]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]]]]]]], "+", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]], ")"]]]]]]]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", "1"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]], "+", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"], ")"]], RowBox[List["Abs", "[", "\[Nu]", "]"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", 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]", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]]]]]], ")"]], " ", 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", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]]]]]]]]], "/;", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]

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

 2007-05-02