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 KelvinKei

 http://functions.wolfram.com/03.19.06.0046.01

 Input Form

 KelvinKei[\[Nu], z] \[Proportional] (1/(8 Sqrt[2 Pi])) ((E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2) (1/Sqrt[(-1)^(3/4) z]) (-4 (-1)^(3/4) Pi - (3 Pi I Sqrt[I z^2])/z + 4 (-Log[z] + Log[(-1)^(3/4) z]) (Sqrt[I z^2]/z)) + E^((I z)/Sqrt[2] + (I Pi \[Nu])/2) (1/Sqrt[(-(-1)^(1/4)) z]) ((-1)^(1/4) Pi + 4 (-1)^(3/4) (-Log[z] + Log[(-(-1)^(1/4)) z])))/ E^(z/Sqrt[2]) + E^(z/Sqrt[2]) (E^((I z)/Sqrt[2] + (I Pi \[Nu])/2) (1/Sqrt[(-1)^(3/4) z]) ((3 Pi)/(-1)^(3/4) - 4 (-1)^(3/4) (-Log[z] + Log[(-1)^(3/4) z])) + E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2) (1/Sqrt[(-(-1)^(1/4)) z]) (4 (-1)^(3/4) Pi + Pi (Sqrt[(-I) z^2]/z) + (4 I Sqrt[(-I) z^2] (-Log[z] + Log[(-(-1)^(1/4)) z]))/z))) (1 + O[1/z]) /; (Abs[z] -> Infinity) && Element[\[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox["1", RowBox[List["8", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " "]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", " ", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]]], "+", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], FractionBox["1", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "\[Pi]"]], "-", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]]]]], "z"], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]], "]"]]]], ")"]], FractionBox[SqrtBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]]], "z"]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], FractionBox["1", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "\[Pi]"]], "+", " ", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], FractionBox["1", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]]]], RowBox[List["(", RowBox[List[RowBox[List["3", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "3"]], "/", "4"]]], "\[Pi]"]], "-", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]], "]"]]]], ")"]]]]]], " ", ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]]], "+", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], FractionBox["1", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]]]], RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "\[Pi]"]], "+", RowBox[List["\[Pi]", FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["z", "2"]]]], "z"]]], "+", " ", FractionBox[RowBox[List["4", " ", "\[ImaginaryI]", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["z", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]], "]"]]]], ")"]]]], "z"]]], ")"]]]]]], ")"]]]]]], ")"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]

 MathML Form

 kei ν ( z ) 1 8 2 π ( z 2 ( 3 π ν 2 - z 2 - - 1 4 z ( 4 - z 2 ( log ( - - 1 4 z ) - log ( z ) ) z + π - z 2 z + 4 ( - 1 ) 3 / 4 π ) + z 2 + π ν 2 ( - 1 ) 3 / 4 z ( 3 ( - 1 ) - 3 / 4 π - 4 ( - 1 ) 3 / 4 ( log ( ( - 1 ) 3 / 4 z ) - log ( z ) ) ) ) + - z 2 ( 3 π ν 2 - z 2 ( - 1 ) 3 / 4 z ( - 4 ( - 1 ) 3 / 4 π - 3 π z 2 z + 4 ( log ( ( - 1 ) 3 / 4 z ) - log ( z ) ) z 2 z ) + z 2 + π ν 2 - - 1 4 z ( 4 ( - 1 ) 3 / 4 ( log ( - - 1 4 z ) - log ( z ) ) + - 1 4 π ) ) ) ( 1 + O ( 1 z ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) ν TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition Proportional KelvinKei ν z 1 8 2 1 2 -1 z 2 1 2 -1 3 ν 2 -1 -1 z 2 1 2 -1 -1 -1 1 4 z 1 2 -1 4 -1 z 2 1 2 -1 -1 1 4 z -1 z z -1 -1 z 2 1 2 z -1 4 -1 3 4 z 2 1 2 -1 ν 2 -1 -1 3 4 z 1 2 -1 3 -1 -3 4 -1 4 -1 3 4 -1 3 4 z -1 z -1 z 2 1 2 -1 3 ν 2 -1 -1 z 2 1 2 -1 -1 3 4 z 1 2 -1 -4 -1 3 4 -1 3 z 2 1 2 z -1 4 -1 3 4 z -1 z z 2 1 2 z -1 z 2 1 2 -1 ν 2 -1 -1 -1 1 4 z 1 2 -1 4 -1 3 4 -1 -1 1 4 z -1 z -1 1 4 1 O 1 z -1 Rule z ν [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKei", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]]], "+", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "\[Pi]"]], "-", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]]]]], "z"], "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]], "]"]]]], ")"]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]]]]], "z"]]], ")"]]]], SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "\[Pi]"]], "+", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]], "]"]]]], ")"]]]]]], ")"]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "3"]], "/", "4"]]], " ", "\[Pi]"]], "-", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]], "]"]]]], ")"]]]]]], ")"]]]], SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]]], "+", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "\[Pi]"]], "+", FractionBox[RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["z", "2"]]]]]], "z"], "+", FractionBox[RowBox[List["4", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "z", "]"]]]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]], "]"]]]], ")"]]]], "z"]]], ")"]]]], SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", "z"], "]"]]]], ")"]]]], RowBox[List["8", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]

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

 2007-05-02