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 KelvinBei

 http://functions.wolfram.com/03.13.06.0030.01

 Input Form

 KelvinBei[z] \[Proportional] (-(I/(Sqrt[2 Pi] Sqrt[-z]))) (E^(z/Sqrt[2]) Sin[(1/8) (Pi - 4 Sqrt[2] z)] - (I Sin[(1/8) (Pi + 4 Sqrt[2] z)])/E^(z/Sqrt[2]) + (1/(8 z)) (E^(z/Sqrt[2]) Cos[(1/8) (Pi + 4 Sqrt[2] z)] + (I Cos[(1/8) (-Pi + 4 Sqrt[2] z)])/E^(z/Sqrt[2])) + (9/(128 z^2)) (E^(z/Sqrt[2]) Cos[(1/8) (-Pi + 4 Sqrt[2] z)] - (I Cos[(1/8) (-Pi - 4 Sqrt[2] z)])/E^(z/Sqrt[2])) + (75/(1024 z^3)) ((-E^(z/Sqrt[2])) Sin[(1/8) (-Pi - 4 Sqrt[2] z)] + (I Sin[(1/8) (Pi - 4 Sqrt[2] z)])/E^(z/Sqrt[2])) + \[Ellipsis]) /; (z -> -Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinBei", "[", "z", "]"]], "\[Proportional]", RowBox[List[RowBox[List["-", FractionBox["\[ImaginaryI]", RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox[RowBox[List["-", "z"]]]]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["8", " ", "z"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Pi]"]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["9", " "]], RowBox[List["128", " ", SuperscriptBox["z", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Pi]"]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Pi]"]], "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["75", " "]], RowBox[List["1024", " ", SuperscriptBox["z", "3"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Pi]"]], "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]]]], ")"]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["-", "\[Infinity]"]]]], ")"]]]]]]

 MathML Form

 bei ( z ) - 2 π - z ( z 2 sin ( 1 8 ( π - 4 2 z ) ) - - z 2 sin ( 1 8 ( 4 2 z + π ) ) + 1 8 z ( - z 2 cos ( 1 8 ( 4 2 z - π ) ) + z 2 cos ( 1 8 ( 4 2 z + π ) ) ) + 9 128 z 2 ( z 2 cos ( 1 8 ( 4 2 z - π ) ) - - z 2 cos ( 1 8 ( - 4 2 z - π ) ) ) + 75 1024 z 3 ( - z 2 sin ( 1 8 ( π - 4 2 z ) ) - z 2 sin ( 1 8 ( - 4 2 z - π ) ) ) + ) /; ( z "\[Rule]" - ) Condition Proportional KelvinBei z -1 2 1 2 -1 z 1 2 -1 z 2 1 2 -1 1 8 -1 4 2 1 2 z -1 -1 z 2 1 2 -1 1 8 4 2 1 2 z 1 8 z -1 -1 z 2 1 2 -1 1 8 4 2 1 2 z -1 z 2 1 2 -1 1 8 4 2 1 2 z 9 128 z 2 -1 z 2 1 2 -1 1 8 4 2 1 2 z -1 -1 -1 z 2 1 2 -1 1 8 -4 2 1 2 z -1 75 1024 z 3 -1 -1 z 2 1 2 -1 1 8 -1 4 2 1 2 z -1 z 2 1 2 -1 1 8 -4 2 1 2 z -1 Rule z -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinBei", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Pi]"]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]]]], RowBox[List["8", " ", "z"]]], "+", FractionBox[RowBox[List["9", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Pi]"]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Pi]"]], "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]]]], ")"]]]], RowBox[List["128", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List["75", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Pi]"]], "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]]]], ")"]]]], RowBox[List["1024", " ", SuperscriptBox["z", "3"]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox[RowBox[List["-", "z"]]]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["-", "\[Infinity]"]]]], ")"]]]]]]]]

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

 2007-05-02