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 KelvinBei

 http://functions.wolfram.com/03.17.06.0044.01

 Input Form

 KelvinBei[\[Nu], z] \[Proportional] (-(1/(2 Sqrt[2 Pi] Sqrt[-z]))) (((E^((I z)/Sqrt[2] + (I Pi \[Nu])/2 - (3 Pi I)/8) (1 + O[1/z^2]) + E^(-((I z)/Sqrt[2]) + (6 I Pi \[Nu])/4 + (3 Pi I)/8) (1 + O[1/z^2]))/ E^(z/Sqrt[2]) + E^(z/Sqrt[2]) ((-E^((I z)/Sqrt[2] + (I Pi \[Nu])/2 - (Pi I)/8)) (1 + O[1/z^2]) + E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2 + (Pi I)/8) (1 + O[1/z^2]))) + ((1 - \[Nu]^2)/(8 z)) (((-E^((I z)/Sqrt[2] + (I Pi \[Nu])/2 - (Pi I)/8)) (1 + O[1/z^2]) - E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2 + (Pi I)/8) (1 + O[1/z^2]))/E^(z/Sqrt[2]) + E^(z/Sqrt[2]) ((-E^((I z)/Sqrt[2] + (5 I Pi \[Nu])/2 - (3 Pi I)/8)) (1 + O[1/z^2]) + E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2 + (3 Pi I)/8) (1 + O[1/z^2])))) /; Inequality[Pi/2, Less, Arg[z], LessEqual, Pi] && (Abs[z] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox[RowBox[List["-", "z"]]]]]]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", 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[RowBox[List["3", "\[Pi]", " ", "\[ImaginaryI]"]], "8"]]]], 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["6", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "+", FractionBox[RowBox[List["3", "\[Pi]", " ", "\[ImaginaryI]"]], "8"]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", "2"]], "]"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], RowBox[List["(", 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[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "8"]]]]]], 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"], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "8"]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", "2"]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", "+", RowBox[List[FractionBox[RowBox[List["1", "-", SuperscriptBox["\[Nu]", "2"]]], RowBox[List["8", " ", "z"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], RowBox[List["(", 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[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "8"]]]]]], 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"], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "8"]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", "2"]], "]"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]], "+", FractionBox[RowBox[List["5", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"], "-", FractionBox[RowBox[List["3", "\[Pi]", " ", "\[ImaginaryI]"]], "8"]]]]]], 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"], "+", FractionBox[RowBox[List["3", "\[Pi]", " ", "\[ImaginaryI]"]], "8"]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", "2"]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

 MathML Form

 bei ν ( z ) - 1 2 2 π - z ( z 2 ( - z 2 + 3 π ν 2 + π 8 ( 1 + O ( 1 z 2 ) ) TagBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "(", FractionBox["1", SuperscriptBox["z", "2"]], ")"]]]], ")"]], HypergeometricPFQ] - z 2 + π ν 2 - π 8 ( 1 + O ( 1 z 2 ) ) TagBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "(", FractionBox["1", SuperscriptBox["z", "2"]], ")"]]]], ")"]], HypergeometricPFQ] ) + - z 2 ( z 2 + π ν 2 - 3 π 8 ( 1 + O ( 1 z 2 ) ) TagBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "(", FractionBox["1", SuperscriptBox["z", "2"]], ")"]]]], ")"]], HypergeometricPFQ] + - z 2 + 6 π ν 4 + 3 π 8 ( 1 + O ( 1 z 2 ) ) TagBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "(", FractionBox["1", SuperscriptBox["z", "2"]], ")"]]]], ")"]], HypergeometricPFQ] ) + 1 - ν 2 8 z ( z 2 ( - z 2 + 3 π ν 2 + 3 π 8 ( 1 + O ( 1 z 2 ) ) TagBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "(", FractionBox["1", SuperscriptBox["z", "2"]], ")"]]]], ")"]], HypergeometricPFQ] - z 2 + 5 π ν 2 - 3 π 8 ( 1 + O ( 1 z 2 ) ) TagBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "(", FractionBox["1", SuperscriptBox["z", "2"]], ")"]]]], ")"]], HypergeometricPFQ] ) + - z 2 ( - z 2 + π ν 2 - π 8 ( 1 + O ( 1 z 2 ) ) TagBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "(", FractionBox["1", SuperscriptBox["z", "2"]], ")"]]]], ")"]], HypergeometricPFQ] - - z 2 + 3 π ν 2 + π 8 ( 1 + O ( 1 z 2 ) ) TagBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "(", FractionBox["1", SuperscriptBox["z", "2"]], ")"]]]], ")"]], HypergeometricPFQ] ) ) ) /; π 2 < arg ( z ) π ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional KelvinBei ν z -1 1 2 2 1 2 -1 z 1 2 -1 z 2 1 2 -1 -1 z 2 1 2 -1 3 ν 2 -1 8 -1 HypergeometricPFQ 1 O 1 z 2 -1 -1 z 2 1 2 -1 ν 2 -1 -1 8 -1 HypergeometricPFQ 1 O 1 z 2 -1 -1 z 2 1 2 -1 z 2 1 2 -1 ν 2 -1 -1 3 8 -1 HypergeometricPFQ 1 O 1 z 2 -1 -1 z 2 1 2 -1 6 ν 4 -1 3 8 -1 HypergeometricPFQ 1 O 1 z 2 -1 1 -1 ν 2 8 z -1 z 2 1 2 -1 -1 z 2 1 2 -1 3 ν 2 -1 3 8 -1 HypergeometricPFQ 1 O 1 z 2 -1 -1 z 2 1 2 -1 5 ν 2 -1 -1 3 8 -1 HypergeometricPFQ 1 O 1 z 2 -1 -1 z 2 1 2 -1 -1 z 2 1 2 -1 ν 2 -1 -1 8 -1 HypergeometricPFQ 1 O 1 z 2 -1 -1 -1 z 2 1 2 -1 3 ν 2 -1 8 -1 HypergeometricPFQ 1 O 1 z 2 -1 Inequality 2 -1 z Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", 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[RowBox[List["3", " ", "\[Pi]", " ", "\[ImaginaryI]"]], "8"]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "2"]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]]], "+", 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"z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox[RowBox[List["-", "z"]]]]]]]], "/;", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]]]

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

 2007-05-02