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 KelvinBer

 http://functions.wolfram.com/03.18.06.0006.01

 Input Form

 KelvinBer[\[Nu], z] == (2^(-1 - 2 \[Nu]) Sqrt[Pi] Subscript[z, 0]^\[Nu] Gamma[1 + \[Nu]] (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^k/(Subscript[z, 0]^k k!)) (E^((3 I Pi \[Nu])/2) HypergeometricPFQRegularized[{(1 + \[Nu])/2, (2 + \[Nu])/2}, {(1 - k + \[Nu])/2, (2 - k + \[Nu])/2, 1 + \[Nu]}, (I Subscript[z, 0]^2)/4] + HypergeometricPFQRegularized[ {(1 + \[Nu])/2, (2 + \[Nu])/2}, {(1 - k + \[Nu])/2, (2 - k + \[Nu])/2, 1 + \[Nu]}, -((I Subscript[z, 0]^2)/4)]) (z - Subscript[z, 0])^k, {k, 0, Infinity}])/E^((3/4) I Pi \[Nu])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "\[Nu]"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["3", "4"]]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", SubsuperscriptBox["z", "0", "\[Nu]"], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], 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", "k"], SubsuperscriptBox["z", "0", RowBox[List["-", "k"]]]]], RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "k", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "-", "k", "+", "\[Nu]"]], "2"], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SubsuperscriptBox["z", "0", "2"]]], "4"]]], "]"]]]], "+", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "k", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "-", "k", "+", "\[Nu]"]], "2"], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SubsuperscriptBox["z", "0", "2"]]], "4"]]]]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]

 MathML Form

 ber ν ( z ) 2 - 2 ν - 1 - 3 π ν 4 π z 0 ν Γ ( ν + 1 ) ( 1 z 0 ) ν arg ( z - z 0 ) 2 π z 0 ν arg ( z - z 0 ) 2 π k = 0 2 k z 0 - k k ! ( 3 π ν 2 2 F ~ 3 ( ν + 1 2 , ν + 2 2 ; 1 - k + ν 2 , 2 - k + ν 2 , ν + 1 ; z 0 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "2"]], "2"], 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[RowBox[List["1", "-", "k", "+", "\[Nu]"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["2", "-", "k", "+", "\[Nu]"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[RowBox[List["\[ImaginaryI]", " ", SubsuperscriptBox["z", "0", "2"]]], "4"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQRegularized] + 2 F ~ 3 ( ν + 1 2 , ν + 2 2 ; 1 - k + ν 2 , 2 - k + ν 2 , ν + 1 ; - z 0 2 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "3"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Nu]", "+", "1"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["\[Nu]", "+", "2"]], "2"], 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[RowBox[List["1", "-", "k", "+", "\[Nu]"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox[RowBox[List["2", "-", "k", "+", "\[Nu]"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["\[Nu]", "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SubsuperscriptBox["z", "0", "2"]]], "4"]]], HypergeometricPFQRegularized, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQRegularized] ) ( z - z 0 ) k KelvinBer ν z 2 -2 ν -1 -1 3 ν 4 -1 1 2 Subscript z 0 ν Gamma ν 1 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 k Subscript z 0 -1 k k -1 3 ν 2 -1 HypergeometricPFQRegularized ν 1 2 -1 ν 2 2 -1 1 -1 k ν 2 -1 2 -1 k ν 2 -1 ν 1 Subscript z 0 2 4 -1 HypergeometricPFQRegularized ν 1 2 -1 ν 2 2 -1 1 -1 k ν 2 -1 2 -1 k ν 2 -1 ν 1 -1 Subscript z 0 2 4 -1 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["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "\[Nu]"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["-", "3"]], ")"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], " ", SqrtBox["\[Pi]"], " ", SubsuperscriptBox["zz", "0", "\[Nu]"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", 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", "k"], " ", SubsuperscriptBox["zz", "0", RowBox[List["-", "k"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "k", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "k", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SubsuperscriptBox["zz", "0", "2"]]], "4"]]], "]"]]]], "+", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "k", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "k", "+", "\[Nu]"]], ")"]]]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SubsuperscriptBox["zz", "0", "2"]]], ")"]]]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]]]]]]

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

 2007-05-02