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 KelvinKei

 http://functions.wolfram.com/03.19.03.0037.01

 Input Form

 KelvinKei[\[Nu], z] == (-(((-1)^(1/8) E^((-(-1)^(1/4)) z - (I Pi \[Nu])/2) Sqrt[Pi/2])/ (2 Sqrt[z]))) (Sum[((-(1/2) + 2 k + Abs[\[Nu]])!/ (I^k z^(2 k) (2^(2 k) (2 k)! (-(1/2) - 2 k + Abs[\[Nu]])!))) ((-1)^(1/4) + E^(I (Sqrt[2] z + Pi (-(1/2) + k + \[Nu])))), {k, 0, Floor[(1/4) (-1 + 2 Abs[\[Nu]])]}] + Sum[(((1/2 + 2 k + Abs[\[Nu]])! z^(-2 k - 1))/ (I^k (2^(2 k + 1) (1 + 2 k)! (-(3/2) - 2 k + Abs[\[Nu]])!))) (1 - (-1)^(3/4) E^(I (Sqrt[2] z + Pi (k + \[Nu])))), {k, 0, Floor[(1/4) (-3 + 2 Abs[\[Nu]])]}]) /; Element[\[Nu] - 1/2, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "8"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], " ", SqrtBox[FractionBox["\[Pi]", "2"]]]], RowBox[List["2", " ", SqrtBox["z"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]]], ")"]]]], "]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]], SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "k"]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]], " "]], RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]]], RowBox[List["(", " ", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["2"], " ", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "k", "+", "\[Nu]"]], ")"]]]]]], ")"]]]]]]], " ", ")"]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]]], ")"]]]], "]"]]], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]], SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "k"]]], SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", "1"]]]]]]], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]]], RowBox[List["(", " ", RowBox[List["1", "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["2"], " ", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["k", "+", "\[Nu]"]], ")"]]]]]], ")"]]]]]]]]], ")"]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]]]]]]

 MathML Form

 kei ν ( z ) - - 1 8 2 z - - 1 4 z - π ν 2 π 2 ( k = 0 1 4 ( 2 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 3 ) ( 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 2 ) ! - k z - 2 k - 1 2 2 k + 1 ( 2 k + 1 ) ! ( - 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 3 2 ) ! ( 1 - ( - 1 ) 3 / 4 ( 2 z + π ( k + ν ) ) ) + k = 0 1 4 ( 2 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 ) ( 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 2 ) ! - k z - 2 k 2 2 k ( 2 k ) ! ( - 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 2 ) ! ( - 1 4 + ( 2 z + π ( k + ν - 1 2 ) ) ) ) /; ν - 1 2 TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition KelvinKei ν z -1 -1 1 8 2 z 1 2 -1 -1 -1 1 4 z -1 ν 2 -1 2 -1 1 2 k 0 1 4 2 ν -3 2 k ν 1 2 -1 k z -2 k -1 2 2 k 1 2 k 1 -2 k ν -1 3 2 -1 1 -1 -1 3 4 2 1 2 z k ν k 0 1 4 2 ν -1 2 k ν -1 1 2 -1 k z -2 k 2 2 k 2 k -2 k ν -1 1 2 -1 -1 1 4 2 1 2 z k ν -1 1 2 ν -1 1 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "8"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], " ", SqrtBox[FractionBox["\[Pi]", "2"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]], " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "k"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["2"], " ", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "k", "+", "\[Nu]"]], ")"]]]]]], ")"]]]]]]], ")"]]]], RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]]], ")"]]]], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]], " ", SuperscriptBox["\[ImaginaryI]", RowBox[List["-", "k"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["2"], " ", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["k", "+", "\[Nu]"]], ")"]]]]]], ")"]]]]]]]]], ")"]]]], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["z"]]]]]], "/;", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]]]]]]]]

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

 2007-05-02