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 KelvinBer

 http://functions.wolfram.com/03.18.06.0037.01

 Input Form

 KelvinBer[\[Nu], z] \[Proportional] (1/(Sqrt[2 Pi] Sqrt[z])) (E^(z/Sqrt[2]) Cos[(1/8) (-4 Sqrt[2] z + Pi (1 - 4 \[Nu]))] + I E^(-(z/Sqrt[2]) + I Pi \[Nu]) Sin[(1/8) (-4 Sqrt[2] z + Pi (3 + 4 \[Nu]))] + ((1 - 4 \[Nu]^2)/(8 z)) (I E^(-(z/Sqrt[2]) + I Pi \[Nu]) Sin[(1/8) (4 Sqrt[2] z - Pi (1 + 4 \[Nu]))] + E^(z/Sqrt[2]) Sin[(1/8) (4 Sqrt[2] z + Pi (1 + 4 \[Nu]))]) - ((9 - 40 \[Nu]^2 + 16 \[Nu]^4)/(128 z^2)) (I E^(-(z/Sqrt[2]) + I Pi \[Nu]) Cos[(1/8) (4 Sqrt[2] z - Pi (3 + 4 \[Nu]))] - E^(z/Sqrt[2]) Sin[(1/8) (4 Sqrt[2] z - Pi (1 - 4 \[Nu]))]) - ((225 - 1036 \[Nu]^2 + 560 \[Nu]^4 - 64 \[Nu]^6)/(3072 z^3)) (E^(z/Sqrt[2]) Cos[(1/8) (-4 Sqrt[2] z - Pi (1 + 4 \[Nu]))] - I E^(-(z/Sqrt[2]) + I Pi \[Nu]) Cos[(1/8) (-4 Sqrt[2] z + Pi (1 + 4 \[Nu]))]) + \[Ellipsis]) /; Inequality[-(Pi/2), Less, Arg[z], LessEqual, Pi] && (Abs[z] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox["1", RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], SqrtBox["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[RowBox[List[RowBox[List["-", "4"]], " ", SqrtBox["2"], " ", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["z", SqrtBox["2"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SqrtBox["2"], " ", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["1", "-", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]]]], RowBox[List["8", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["z", SqrtBox["2"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["2"], " ", "z"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["2"], " ", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List["9", "-", RowBox[List["40", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["16", " ", SuperscriptBox["\[Nu]", "4"]]]]], RowBox[List["128", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["z", SqrtBox["2"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["2"], " ", "z"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["2"], " ", "z"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List["225", "-", RowBox[List["1036", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["560", " ", SuperscriptBox["\[Nu]", "4"]]], "-", RowBox[List["64", " ", SuperscriptBox["\[Nu]", "6"]]]]], RowBox[List["3072", " ", SuperscriptBox["z", "3"]]]], " ", 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[RowBox[List["-", "4"]], " ", SqrtBox["2"], " ", "z"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["z", SqrtBox["2"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SqrtBox["2"], " ", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]]]], ")"]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[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

 ber ν ( z ) 1 2 π z ( z 2 cos ( 1 8 ( π ( 1 - 4 ν ) - 4 2 z ) ) + π ν - z 2 sin ( 1 8 ( π ( 4 ν + 3 ) - 4 2 z ) ) + 1 - 4 ν 2 8 z ( z 2 sin ( 1 8 ( 4 2 z + π ( 4 ν + 1 ) ) ) + π ν - z 2 sin ( 1 8 ( 4 2 z - π ( 4 ν + 1 ) ) ) ) - 16 ν 4 - 40 ν 2 + 9 128 z 2 ( π ν - z 2 cos ( 1 8 ( 4 2 z - π ( 4 ν + 3 ) ) ) - z 2 sin ( 1 8 ( 4 2 z - π ( 1 - 4 ν ) ) ) ) - - 64 ν 6 + 560 ν 4 - 1036 ν 2 + 225 3072 z 3 ( z 2 cos ( 1 8 ( - 4 2 z - π ( 4 ν + 1 ) ) ) - π ν - z 2 cos ( 1 8 ( π ( 4 ν + 1 ) - 4 2 z ) ) ) + ) /; - π 2 < arg ( z ) π ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional KelvinBer ν z 1 2 1 2 z 1 2 -1 z 2 1 2 -1 1 8 1 -1 4 ν -1 4 2 1 2 z ν -1 z 2 1 2 -1 1 8 4 ν 3 -1 4 2 1 2 z 1 -1 4 ν 2 8 z -1 z 2 1 2 -1 1 8 4 2 1 2 z 4 ν 1 ν -1 z 2 1 2 -1 1 8 4 2 1 2 z -1 4 ν 1 -1 16 ν 4 -1 40 ν 2 9 128 z 2 -1 ν -1 z 2 1 2 -1 1 8 4 2 1 2 z -1 4 ν 3 -1 z 2 1 2 -1 1 8 4 2 1 2 z -1 1 -1 4 ν -1 -64 ν 6 560 ν 4 -1 1036 ν 2 225 3072 z 3 -1 z 2 1 2 -1 1 8 -4 2 1 2 z -1 4 ν 1 -1 ν -1 z 2 1 2 -1 cos 1 8 4 ν 1 -1 4 2 1 2 z Inequality -1 2 -1 z Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SqrtBox["2"], " ", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["z", SqrtBox["2"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SqrtBox["2"], " ", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["z", SqrtBox["2"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["2"], " ", "z"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["2"], " ", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]]]], ")"]]]], RowBox[List["8", " ", "z"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["9", "-", RowBox[List["40", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["16", " ", SuperscriptBox["\[Nu]", "4"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["z", SqrtBox["2"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["2"], " ", "z"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["2"], " ", "z"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]]]], ")"]]]], RowBox[List["128", " ", SuperscriptBox["z", "2"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["225", "-", RowBox[List["1036", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["560", " ", SuperscriptBox["\[Nu]", "4"]]], "-", RowBox[List["64", " ", SuperscriptBox["\[Nu]", "6"]]]]], ")"]], " ", 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[RowBox[List["-", "4"]], " ", SqrtBox["2"], " ", "z"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["z", SqrtBox["2"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SqrtBox["2"], " ", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]]]], ")"]]]], RowBox[List["3072", " ", SuperscriptBox["z", "3"]]]], "+", "\[Ellipsis]"]], RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox["z"]]]], "/;", RowBox[List[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