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 KelvinKei

 http://functions.wolfram.com/03.15.06.0004.01

 Input Form

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

 Standard Form

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 MathML Form

 kei ( 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 ( z 0 ) - 2 ( - 1 ) k π arg ( z - z 0 ) 2 π arg ( z 0 ) + π 2 π bei k - 4 j ( z 0 ) ) - ( 1 - k ) ( ker 4 j - k ( z 0 ) - 2 ( - 1 ) k π arg ( z - z 0 ) 2 π arg ( z 0 ) + π 2 π ber k - 4 j ( z 0 ) ) ) - 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 ( z 0 ) - 2 ( - 1 ) k π arg ( z - z 0 ) 2 π arg ( z 0 ) + π 2 π bei - 4 j + k - 2 ( z 0 ) ) - ( 1 - k ) ( ker 4 j - k + 2 ( z 0 ) - 2 ( - 1 ) k π arg ( z - z 0 ) 2 π arg ( z 0 ) + π 2 π ber - 4 j + k - 2 ( z 0 ) ) ) ) ( z - z 0 ) k KelvinKei 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 k KelvinKei 4 j -1 k Subscript z 0 -1 2 -1 k z -1 Subscript z 0 2 -1 Subscript z 0 2 -1 KelvinBei k -1 4 j Subscript z 0 -1 1 -1 k KelvinKer 4 j -1 k Subscript z 0 -1 2 -1 k z -1 Subscript z 0 2 -1 Subscript z 0 2 -1 KelvinBer k -1 4 j Subscript z 0 -1 j 0 k -1 2 -1 Binomial k 2 j 1 1 k KelvinKei 4 j -1 k 2 Subscript z 0 -1 2 -1 k z -1 Subscript z 0 2 -1 Subscript z 0 2 -1 KelvinBei -4 j k -2 Subscript z 0 -1 1 -1 k KelvinKer 4 j -1 k 2 Subscript z 0 -1 2 -1 k z -1 Subscript z 0 2 -1 Subscript z 0 2 -1 KelvinBer -4 j k -2 Subscript z 0 z -1 Subscript z 0 k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKei", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", 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[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["(", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02