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 KelvinKei

 http://functions.wolfram.com/03.15.20.0006.01

 Input Form

 D[KelvinKei[z], {z, \[Alpha]}] == ((I z^(2 - \[Alpha]))/4) Sum[((-1)^k FDLogConstant[z, 4 k + 2, \[Alpha]] z^(4 k))/(2^(4 k) (1 + 2 k)!^2), {k, 0, Infinity}] - (Pi/(z^\[Alpha] 4)) Sum[((-1)^k (4 k)! z^(4 k))/ (2^(4 k) ((2 k)!^2 Gamma[1 + 4 k - \[Alpha]])), {k, 0, Infinity}] + I z^(2 - \[Alpha]) Sum[(((-1)^k (4 k + 2)! (Log[2] + PolyGamma[2 + 2 k]))/ (2^(4 k) ((1 + 2 k)!^2 Gamma[3 + 4 k - \[Alpha]]))) z^(4 k), {k, 0, Infinity}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["KelvinKei", "[", "z", "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", RowBox[List["2", "-", "\[Alpha]"]]]]], "4"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox["2", RowBox[List[RowBox[List["-", "4"]], " ", "k"]]], " ", RowBox[List["FDLogConstant", "[", RowBox[List["z", ",", RowBox[List[RowBox[List["4", "k"]], "+", "2"]], ",", "\[Alpha]"]], "]"]], SuperscriptBox["z", RowBox[List["4", "k"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], ")"]], "2"]]]]]], "-", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], "4"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox["2", RowBox[List[RowBox[List["-", "4"]], " ", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["4", "k"]], ")"]], "!"]], SuperscriptBox["z", RowBox[List["4", " ", "k"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]], ")"]], "2"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", RowBox[List["4", " ", "k"]], "-", "\[Alpha]"]], "]"]]]]]]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", RowBox[List["2", "-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "4"]], " ", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", "k"]], "+", "2"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "2", "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "k"]]]], "]"]]]], ")"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], ")"]], "2"], " ", RowBox[List["Gamma", "[", RowBox[List["3", "+", RowBox[List["4", " ", "k"]], "-", "\[Alpha]"]], "]"]]]]], SuperscriptBox["z", RowBox[List["4", "k"]]]]]]]]]]]]]]]

 MathML Form

 α kei ( z ) z α z 2 - α k = 0 ( - 1 ) k 2 - 4 k ( 4 k + 2 ) ! ( log ( 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( 2 k + 2 ) ) ( ( 2 k + 1 ) ! ) 2 Γ ( 4 k - α + 3 ) z 4 k + z 2 - α 4 k = 0 ( - 1 ) k 2 - 4 k ℱ𝒞 log ( α ) ( z , 4 k + 2 ) ( ( 2 k + 1 ) ! ) 2 z 4 k - π z - α 4 k = 0 ( - 1 ) k 2 - 4 k ( 4 k ) ! z 4 k ( ( 2 k ) ! ) 2 Γ ( 4 k - α + 1 ) z α KelvinKei z z 2 -1 α k 0 -1 k 2 -4 k 4 k 2 2 PolyGamma 2 k 2 2 k 1 2 Gamma 4 k -1 α 3 -1 z 4 k z 2 -1 α 4 -1 k 0 -1 k 2 -4 k Subscript ℱ𝒞 log α z 4 k 2 2 k 1 2 -1 z 4 k -1 z -1 α 4 -1 k 0 -1 k 2 -4 k 4 k z 4 k 2 k 2 Gamma 4 k -1 α 1 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["KelvinKei", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", RowBox[List["2", "-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "4"]], " ", "k"]]], " ", RowBox[List["FDLogConstant", "[", RowBox[List["z", ",", RowBox[List[RowBox[List["4", " ", "k"]], "+", "2"]], ",", "\[Alpha]"]], "]"]], " ", SuperscriptBox["z", RowBox[List["4", " ", "k"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], ")"]], "2"]]]]]], "-", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "4"]], " ", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["4", " ", "k"]], ")"]], "!"]], " ", SuperscriptBox["z", RowBox[List["4", " ", "k"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]], ")"]], "2"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", RowBox[List["4", " ", "k"]], "-", "\[Alpha]"]], "]"]]]]]]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", RowBox[List["2", "-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "4"]], " ", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", "2"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "2", "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", "+", RowBox[List["2", " ", "k"]]]], "]"]]]], ")"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["4", " ", "k"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], ")"]], "2"], " ", RowBox[List["Gamma", "[", RowBox[List["3", "+", RowBox[List["4", " ", "k"]], "-", "\[Alpha]"]], "]"]]]]]]]]]]]]]]]

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

 2007-05-02