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 KelvinKer

 http://functions.wolfram.com/03.20.06.0007.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", "\n", "\t", RowBox[List[FractionBox["1", "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "3"]], RowBox[List["k", "/", "2"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", "-", "1"]], ")"]], "k"]]], RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["k", "/", "2"]], "]"]]], " ", RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", RowBox[List["2", "j"]]]], "]"]], RowBox[List["(", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", SubscriptBox["z", "0"], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "+", "k", "-", RowBox[List["4", " ", "j"]]]], ",", SubscriptBox["z", "0"]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", SubscriptBox["z", "0"]], ")"]], RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SubsuperscriptBox["z", "0", RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List["\[Nu]", "-", "k", "+", RowBox[List["4", " ", "j"]]]], ",", SubscriptBox["z", "0"]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"]]], "2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", SubscriptBox["z", "0"], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "+", "k", "-", RowBox[List["4", " ", "j"]]]], ",", SubscriptBox["z", "0"]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", SubscriptBox["z", "0"]], ")"]], RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SubsuperscriptBox["z", "0", RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List["\[Nu]", "-", "k", "+", RowBox[List["4", " ", "j"]]]], ",", SubscriptBox["z", "0"]]], "]"]]]]]], ")"]]]]]], ")"]], ")"]]]]]], "-", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]], "/", "2"]], "]"]]], " ", RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["2", "j"]], "+", "1"]]]], "]"]], RowBox[List["(", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"]]], "2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", SubscriptBox["z", "0"], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "+", "k", "-", RowBox[List["4", " ", "j"]], "-", "2"]], ",", SubscriptBox["z", "0"]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", SubscriptBox["z", "0"]], ")"]], RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SubsuperscriptBox["z", "0", RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List["\[Nu]", "-", "k", "+", RowBox[List["4", " ", "j"]], "+", "2"]], ",", SubscriptBox["z", "0"]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"]]], "2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", SubscriptBox["z", "0"], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "+", "k", "-", RowBox[List["4", " ", "j"]], "-", "2"]], ",", SubscriptBox["z", "0"]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", SubscriptBox["z", "0"]], ")"]], RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SubsuperscriptBox["z", "0", RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], RowBox[List["KelvinKer", "[", RowBox[List[RowBox[List["\[Nu]", "-", "k", "+", RowBox[List["4", " ", "j"]], "+", "2"]], ",", SubscriptBox["z", "0"]]], "]"]]]]]], ")"]]]]]], ")"]], ")"]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]

 MathML Form

 ker ν ( z ) 1 2 k = 0 2 - 3 k 2 ( - 1 ) k 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 ) ( ( 1 z 0 ) ν arg ( z - z 0 ) 2 π z 0 ν arg ( z - z 0 ) 2 π kei 4 j - k + ν ( z 0 ) - 2 π ( - 1 ) k cos ( π ν ) arg ( z - z 0 ) 2 π arg ( z 0 ) + π 2 π bei - 4 j + k - ν ( z 0 ) ) + ( 1 + k ) ( ( 1 z 0 ) ν arg ( z - z 0 ) 2 π z 0 ν arg ( z - z 0 ) 2 π ker 4 j - k + ν ( z 0 ) - ( - 1 ) k 2 π cos ( π ν ) arg ( z - z 0 ) 2 π arg ( z 0 ) + π 2 π ber - 4 j + k - ν ( 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 ) ( ( 1 z 0 ) ν arg ( z - z 0 ) 2 π z 0 ν arg ( z - z 0 ) 2 π kei 4 j - k + ν + 2 ( z 0 ) - ( - 1 ) k 2 π cos ( π ν ) arg ( z - z 0 ) 2 π arg ( z 0 ) + π 2 π bei - 4 j + k - ν - 2 ( z 0 ) ) + ( 1 + k ) ( ( 1 z 0 ) ν arg ( z - z 0 ) 2 π z 0 ν arg ( z - z 0 ) 2 π ker 4 j - k + ν + 2 ( z 0 ) - ( - 1 ) k 2 π cos ( π ν ) arg ( z - z 0 ) 2 π arg ( z 0 ) + π 2 π ber - 4 j + k - ν - 2 ( z 0 ) ) ) ) ( z - z 0 ) k KelvinKer ν z 1 2 k 0 2 -1 3 k 2 -1 -1 k k -1 j 0 k 2 -1 Binomial k 2 j 1 -1 k 1 Subscript z 0 -1 ν z -1 Subscript z 0 2 -1 Subscript z 0 ν z -1 Subscript z 0 2 -1 KelvinKei 4 j -1 k ν Subscript z 0 -1 2 -1 k ν z -1 Subscript z 0 2 -1 Subscript z 0 2 -1 KelvinBei -4 j k -1 ν Subscript z 0 1 k 1 Subscript z 0 -1 ν z -1 Subscript z 0 2 -1 Subscript z 0 ν z -1 Subscript z 0 2 -1 KelvinKer 4 j -1 k ν Subscript z 0 -1 -1 k 2 ν z -1 Subscript z 0 2 -1 Subscript z 0 2 -1 KelvinBer -4 j k -1 ν Subscript z 0 -1 j 0 k -1 2 -1 Binomial k 2 j 1 1 -1 k 1 Subscript z 0 -1 ν z -1 Subscript z 0 2 -1 Subscript z 0 ν z -1 Subscript z 0 2 -1 KelvinKei 4 j -1 k ν 2 Subscript z 0 -1 -1 k 2 ν z -1 Subscript z 0 2 -1 Subscript z 0 2 -1 KelvinBei -4 j k -1 ν -2 Subscript z 0 1 k 1 Subscript z 0 -1 ν z -1 Subscript z 0 2 -1 Subscript z 0 ν z -1 Subscript z 0 2 -1 KelvinKer 4 j -1 k ν 2 Subscript z 0 -1 -1 k 2 ν z -1 Subscript z 0 2 -1 Subscript z 0 2 -1 KelvinBer -4 j k -1 ν -2 Subscript z 0 z -1 Subscript z 0 k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", FractionBox[RowBox[List["3", " ", "k"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", "-", "1"]], ")"]], "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["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", 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SubsuperscriptBox["zz", "0", RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List["\[Nu]", "-", "k", "+", RowBox[List["4", " ", "j"]]]], ",", SubscriptBox["zz", "0"]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"]]], " ", "2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", 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RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "+", "k", "-", RowBox[List["4", " ", "j"]], "-", "2"]], ",", SubscriptBox["zz", "0"]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", SubscriptBox["zz", "0"]], ")"]], RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SubsuperscriptBox["zz", "0", RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["KelvinKei", "[", RowBox[List[RowBox[List["\[Nu]", "-", "k", "+", RowBox[List["4", " ", "j"]], "+", "2"]], ",", SubscriptBox["zz", "0"]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02