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 KelvinKei

 http://functions.wolfram.com/03.15.20.0007.01

 Input Form

 D[KelvinKei[z], {z, \[Alpha]}] == ((-2^(-(3/2) + 2 \[Alpha])) Pi^3 HypergeometricPFQRegularized[{1/4, 3/4}, {1/2, (1 - \[Alpha])/4, (2 - \[Alpha])/4, (3 - \[Alpha])/4, 1 - \[Alpha]/4}, -(z^4/256)])/z^\[Alpha] + I 2^(-(7/2) + 2 \[Alpha]) Pi^2 z^(2 - \[Alpha]) Log[2] HypergeometricPFQRegularized[{3/4, 5/4}, {3/2, 3/4 - \[Alpha]/4, 1 - \[Alpha]/4, 5/4 - \[Alpha]/4, 3/2 - \[Alpha]/4}, -(z^4/256)] + ((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}] + I z^(2 - \[Alpha]) Sum[(((-1)^k (4 k + 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[RowBox[List["-", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "+", RowBox[List["2", " ", "\[Alpha]"]]]]]]], " ", SuperscriptBox["\[Pi]", "3"], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox[RowBox[List["1", "-", "\[Alpha]"]], "4"], ",", FractionBox[RowBox[List["2", "-", "\[Alpha]"]], "4"], ",", FractionBox[RowBox[List["3", "-", "\[Alpha]"]], "4"], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "4"]]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "256"]]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["7", "2"]]], "+", RowBox[List["2", " ", "\[Alpha]"]]]]], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", RowBox[List["2", "-", "\[Alpha]"]]], " ", RowBox[List["Log", "[", "2", "]"]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", FractionBox["5", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["3", "2"], "-", FractionBox["\[Alpha]", "4"]]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "256"]]]]], "]"]]]], "+", 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["\[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["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 2 2 α - 7 2 π 2 log ( 2 ) z 2 - α 2 F ~ 5 ( 3 4 , 5 4 ; 3 2 , 3 4 - α 4 , 1 - α 4 , 5 4 - α 4 , 3 2 - α 4 ; - z 4 256 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "5"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["5", "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Alpha]", "4"]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", "4"]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Alpha]", "4"]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["3", "2"], "-", FractionBox["\[Alpha]", "4"]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "256"]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQRegularized] - 2 2 α - 3 2 π 3 z - α 2 F ~ 5 ( 1 4 , 3 4 ; 1 2 , 1 - α 4 , 2 - α 4 , 3 - α 4 , 1 - α 4 ; - z 4 256 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "5"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["3", "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "\[Alpha]"]], "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["2", "-", "\[Alpha]"]], "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["3", "-", "\[Alpha]"]], "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", "4"]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "256"]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQRegularized] + z 2 - α k = 0 ( - 1 ) k 2 - 4 k ( 4 k + 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 z α KelvinKei z 2 2 α -1 7 2 2 log 2 z 2 -1 α HypergeometricPFQRegularized 3 4 5 4 3 2 3 4 -1 α 4 -1 1 -1 α 4 -1 5 4 -1 α 4 -1 3 2 -1 α 4 -1 -1 z 4 256 -1 -1 2 2 α -1 3 2 3 z -1 α HypergeometricPFQRegularized 1 4 3 4 1 2 1 -1 α 4 -1 2 -1 α 4 -1 3 -1 α 4 -1 1 -1 α 4 -1 -1 z 4 256 -1 z 2 -1 α k 0 -1 k 2 -4 k 4 k 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 [/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[RowBox[List["-", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "+", RowBox[List["2", " ", "\[Alpha]"]]]]]]], " ", SuperscriptBox["\[Pi]", "3"], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox[RowBox[List["1", "-", "\[Alpha]"]], "4"], ",", FractionBox[RowBox[List["2", "-", "\[Alpha]"]], "4"], ",", FractionBox[RowBox[List["3", "-", "\[Alpha]"]], "4"], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "4"]]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "256"]]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["7", "2"]]], "+", RowBox[List["2", " ", "\[Alpha]"]]]]], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", RowBox[List["2", "-", "\[Alpha]"]]], " ", RowBox[List["Log", "[", "2", "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", FractionBox["5", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List["1", "-", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["5", "4"], "-", FractionBox["\[Alpha]", "4"]]], ",", RowBox[List[FractionBox["3", "2"], "-", FractionBox["\[Alpha]", "4"]]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], "256"]]]]], "]"]]]], "+", 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["\[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["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