html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 KelvinKer

 http://functions.wolfram.com/03.20.06.0048.01

 Input Form

 KelvinKer[\[Nu], z] \[Proportional] (1/4) Sqrt[Pi/2] (E^(z/Sqrt[2]) (((-1)^(1/4) E^((I Pi \[Nu])/4) Csc[Pi \[Nu]] (E^((I z)/Sqrt[2] - (3 I Pi \[Nu])/2) ((-1)^(3/4) z)^(-(1/2) + \[Nu]) (1 + O[1/z^2]) - (((-(-1)^(1/4)) z)^(-(1/2) + \[Nu]) (((-1)^(3/4) Sqrt[(-I) z^2] Cos[Pi \[Nu]])/z - Sin[Pi \[Nu]]) (1 + O[1/z^2]))/E^((I z)/Sqrt[2])))/z^\[Nu] + ((-1)^(1/4) z^\[Nu] (E^((I z)/Sqrt[2] + (3 I Pi \[Nu])/2) ((-1)^(3/4) z)^(-(1/2) - \[Nu]) (I - Cot[Pi \[Nu]]) (1 + O[1/z^2]) + (((-(-1)^(1/4)) z)^(-(1/2) - \[Nu]) (I + Cot[Pi \[Nu]]) (((-1)^(3/4) Sqrt[(-I) z^2] Cos[Pi \[Nu]])/z + Sin[Pi \[Nu]]) (1 + O[1/z^2]))/E^((I z)/Sqrt[2])))/ E^((I Pi \[Nu])/4)) + (((-1)^(1/4) E^((I Pi \[Nu])/4) Csc[Pi \[Nu]] (E^((I z)/Sqrt[2]) ((-(-1)^(1/4)) z)^(-(1/2) + \[Nu]) (1 + O[1/z^2]) + E^(-((I z)/Sqrt[2]) - (3 I Pi \[Nu])/2) ((-1)^(3/4) z)^(-(1/2) + \[Nu]) (((-1)^(1/4) Sqrt[I z^2] Cos[Pi \[Nu]])/z + Sin[Pi \[Nu]]) (1 + O[1/z^2])))/z^\[Nu] + ((-1)^(1/4) z^\[Nu] ((-E^((I z)/Sqrt[2])) ((-(-1)^(1/4)) z)^ (-(1/2) - \[Nu]) (I + Cot[Pi \[Nu]]) (1 + O[1/z^2]) + E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2) ((-1)^(3/4) z)^ (-(1/2) - \[Nu]) (I - Cot[Pi \[Nu]]) (((-1)^(1/4) Sqrt[I z^2] Cos[Pi \[Nu]])/z - Sin[Pi \[Nu]]) (1 + O[1/z^2])))/E^((I Pi \[Nu])/4))/E^(z/Sqrt[2])) /; (Abs[z] -> Infinity) && !Element[\[Nu], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox["1", "4"], " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]], " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]], "-", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", 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FractionBox["1", "2"]]], "-", "\[Nu]"]]], RowBox[List["(", RowBox[List["\[ImaginaryI]", "-", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", "2"]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", 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" ", "z"]], SqrtBox["2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", "2"]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]]], "-", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], 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"/", "4"]]]]], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]], RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", "2"]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]]], "+", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "-", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "z"], "-", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", "2"]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 ker ν ( z ) 1 4 π 2 ( z 2 ( - 1 4 π ν 4 csc ( π ν ) z - ν ( z 2 - 3 π ν 2 ( ( - 1 ) 3 / 4 z ) ν - 1 2 ( 1 + O ( 1 z 2 ) ) - - z 2 ( - - 1 4 z ) ν - 1 2 ( ( - 1 ) 3 / 4 - z 2 cos ( π ν ) z - sin ( π ν ) ) ( 1 + O ( 1 z 2 ) ) ) + - 1 4 - π ν 4 z ν ( - z 2 ( + cot ( π ν ) ) ( ( - 1 ) 3 / 4 - z 2 cos ( π ν ) z + sin ( π ν ) ) ( - - 1 4 z ) - ν - 1 2 ( 1 + O ( 1 z 2 ) ) + z 2 + 3 π ν 2 ( ( - 1 ) 3 / 4 z ) - ν - 1 2 ( - cot ( π ν ) ) ( 1 + O ( 1 z 2 ) ) ) ) + - z 2 ( - 1 4 π ν 4 csc ( π ν ) z - ν ( z 2 ( - - 1 4 z ) ν - 1 2 ( 1 + O ( 1 z 2 ) ) + - z 2 - 3 π ν 2 ( ( - 1 ) 3 / 4 z ) ν - 1 2 ( - 1 4 z 2 cos ( π ν ) z + sin ( π ν ) ) ( 1 + O ( 1 z 2 ) ) ) + - 1 4 - π ν 4 z ν ( 3 π ν 2 - z 2 ( ( - 1 ) 3 / 4 z ) - ν - 1 2 ( - cot ( π ν ) ) ( - 1 4 z 2 cos ( π ν ) z - sin ( π ν ) ) ( 1 + O ( 1 z 2 ) ) - z 2 ( - - 1 4 z ) - ν - 1 2 ( + cot ( π ν ) ) ( 1 + O ( 1 z 2 ) ) ) ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) ν TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Proportional KelvinKer ν z 1 4 2 -1 1 2 z 2 1 2 -1 -1 1 4 ν 4 -1 ν z -1 ν z 2 1 2 -1 -1 3 ν 2 -1 -1 3 4 z ν -1 1 2 1 O 1 z 2 -1 -1 -1 z 2 1 2 -1 -1 -1 1 4 z ν -1 1 2 -1 3 4 -1 z 2 1 2 ν z -1 -1 ν 1 O 1 z 2 -1 -1 1 4 -1 ν 4 -1 z ν -1 z 2 1 2 -1 ν -1 3 4 -1 z 2 1 2 ν z -1 ν -1 -1 1 4 z -1 ν -1 1 2 1 O 1 z 2 -1 z 2 1 2 -1 3 ν 2 -1 -1 3 4 z -1 ν -1 1 2 -1 ν 1 O 1 z 2 -1 -1 z 2 1 2 -1 -1 1 4 ν 4 -1 ν z -1 ν z 2 1 2 -1 -1 -1 1 4 z ν -1 1 2 1 O 1 z 2 -1 -1 z 2 1 2 -1 -1 3 ν 2 -1 -1 3 4 z ν -1 1 2 -1 1 4 z 2 1 2 ν z -1 ν 1 O 1 z 2 -1 -1 1 4 -1 ν 4 -1 z ν 3 ν 2 -1 -1 z 2 1 2 -1 -1 3 4 z -1 ν -1 1 2 -1 ν -1 1 4 z 2 1 2 ν z -1 -1 ν 1 O 1 z 2 -1 -1 z 2 1 2 -1 -1 -1 1 4 z -1 ν -1 1 2 ν 1 O 1 z 2 -1 Rule z ν [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]], " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]], "-", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "2"]], "]"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "z"], "-", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "2"]], "]"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], ")"]]]]], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]], "+", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], " 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02