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 KelvinBer

 http://functions.wolfram.com/03.18.06.0007.01

 Input Form

 KelvinBer[\[Nu], z] == (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)]) Sum[((2^(-1 - 3 (k/2)) (I - 1)^k)/k!) (Sum[Binomial[k, 2 j] (I (1 - I^k) KelvinBei[4 j - k + \[Nu], Subscript[z, 0]] + (1 + I^k) KelvinBer[4 j - k + \[Nu], Subscript[z, 0]]), {j, 0, Floor[k/2]}] + Sum[Binomial[k, 2 j + 1] ((-I) (1 - I^k) KelvinBei[2 + 4 j - k + \[Nu], Subscript[z, 0]] - (1 + I^k) KelvinBer[2 + 4 j - k + \[Nu], 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["KelvinBer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", 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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", 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["KelvinBei", "[", RowBox[List[RowBox[List[RowBox[List["4", " ", "j"]], "-", "k", "+", "\[Nu]"]], ",", SubscriptBox["z", "0"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List["4", " ", "j"]], "-", "k", "+", "\[Nu]"]], ",", 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[RowBox[List["-", "\[ImaginaryI]"]], RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["4", " ", "j"]], "-", "k", "+", "\[Nu]"]], ",", SubscriptBox["z", "0"]]], "]"]]]], "-", " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["4", " ", "j"]], "-", "k", "+", "\[Nu]"]], ",", SubscriptBox["z", "0"]]], "]"]]]]]], ")"]], ")"]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", 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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", 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["KelvinBei", "[", RowBox[List[RowBox[List[RowBox[List["4", " ", "j"]], "-", "k", "+", "\[Nu]"]], ",", SubscriptBox["zz", "0"]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List["4", " ", "j"]], "-", "k", "+", "\[Nu]"]], ",", SubscriptBox["zz", "0"]]], "]"]]]]]], ")"]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox[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["KelvinBei", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["4", " ", "j"]], "-", "k", "+", "\[Nu]"]], ",", SubscriptBox["zz", "0"]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["\[ImaginaryI]", "k"]]], ")"]], " ", RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List["2", "+", RowBox[List["4", " ", "j"]], "-", "k", "+", "\[Nu]"]], ",", SubscriptBox["zz", "0"]]], "]"]]]]]], ")"]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]]]]]]

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

 2007-05-02