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 KelvinKei

 http://functions.wolfram.com/03.19.20.0019.01

 Input Form

 D[KelvinKei[\[Nu], z], {z, \[Alpha]}] == (-Pi) 2^(-Abs[\[Nu]] - 2) z^(Abs[\[Nu]] - \[Alpha]) Sum[((Cos[(Pi (2 (k + \[Nu]) + Abs[\[Nu]]))/4] Gamma[2 k + Abs[\[Nu]] + 1])/(k! (k + Abs[\[Nu]])! Gamma[2 k + Abs[\[Nu]] - \[Alpha] + 1])) (z/2)^(2 k), {k, 0, Infinity}] + (-1)^(Abs[\[Nu]] - 1) 2^(Abs[\[Nu]] - 2) z^(-Abs[\[Nu]] - \[Alpha]) Sum[(((E^((I Pi (2 \[Nu] + Abs[\[Nu]]))/4) - (-1)^k/E^((I Pi (2 \[Nu] + Abs[\[Nu]]))/4)) (Abs[\[Nu]] - k - 1)!)/ (k! (-2 k + Abs[\[Nu]] - 1)! Gamma[2 k - Abs[\[Nu]] - \[Alpha] + 1])) (Log[z] - PolyGamma[2 k - Abs[\[Nu]] - \[Alpha] + 1] + PolyGamma[-2 k + Abs[\[Nu]]]) ((I z^2)/4)^k, {k, 0, Floor[(Abs[\[Nu]] - 1)/2]}] + 2^(Abs[\[Nu]] - 2) z^(-Abs[\[Nu]] - \[Alpha]) Sum[(((E^((I Pi (2 \[Nu] + Abs[\[Nu]]))/4) - (-1)^k/E^((I Pi (2 \[Nu] + Abs[\[Nu]]))/4)) (Abs[\[Nu]] - k - 1)! Gamma[2 k - Abs[\[Nu]] + 1])/(k! Gamma[2 k - Abs[\[Nu]] - \[Alpha] + 1])) ((I z^2)/4)^k, {k, Floor[(Abs[\[Nu]] - 1)/2] + 1, Abs[\[Nu]] - 1}] + I^(\[Nu] + Abs[\[Nu]]) 2^(-1 - Abs[\[Nu]]) z^(Abs[\[Nu]] - \[Alpha]) Sum[((E^(-((I Pi Abs[\[Nu]])/4)) - (-1)^k E^((I Pi Abs[\[Nu]])/4))/ (k! (k + Abs[\[Nu]])!)) FDLogConstant[z, 2 k + Abs[\[Nu]], \[Alpha]] ((I z^2)/4)^k, {k, 0, Infinity}] + I^(\[Nu] + Abs[\[Nu]]) 2^(-2 - Abs[\[Nu]]) z^(Abs[\[Nu]] - \[Alpha]) Sum[(((E^(-((I Pi Abs[\[Nu]])/4)) - (-1)^k E^((I Pi Abs[\[Nu]])/4)) Gamma[2 k + Abs[\[Nu]] + 1])/(k! (k + Abs[\[Nu]])! Gamma[2 k + Abs[\[Nu]] - \[Alpha] + 1])) (2 Log[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[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", "\[Pi]"]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "2"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[" ", 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["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", "1"]], "]"]]]]]], 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"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["FDLogConstant", "[", RowBox[List["z", ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ",", "\[Alpha]"]], "]"]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], "4"], ")"]], "k"]]]]]]], "+", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["\[Nu]", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", 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["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", "1"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Alpha]", "+", "1"]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", RowBox[List["Log", "[", "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

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

 Rule Form

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

 2007-05-02