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 KelvinBei

 http://functions.wolfram.com/03.13.06.0023.01

 Input Form

 KelvinBei[z] \[Proportional] (1/(Sqrt[2 Pi] Sqrt[z])) ((1/(2 z)) Sum[((1/(1 + 2 k)!) (-(1/z^2))^k Pochhammer[1/2, 1 + 2 k]^2 ((I (-1)^k Cos[(k Pi)/2 + (1/8) (-Pi + 4 Sqrt[2] z)])/E^(z/Sqrt[2]) - E^(z/Sqrt[2]) Cos[(k Pi)/2 + (1/8) (Pi + 4 Sqrt[2] z)]))/4^k, {k, 0, Floor[(1/2) (-1 + n)]}] - Sum[((1/(2 k)!) (-(1/z^2))^k Pochhammer[1/2, 2 k]^2 ((I Cos[(k Pi)/2 + (1/8) (3 Pi - 4 Sqrt[2] z)])/E^(z/Sqrt[2]) + (-1)^k E^(z/Sqrt[2]) Sin[(k Pi)/2 + (1/8) (Pi - 4 Sqrt[2] z)]))/4^k, {k, 0, Floor[n/2]}] + \[Ellipsis]) /; Inequality[-(Pi/2), Less, Arg[z], LessEqual, Pi] && (Abs[z] -> Infinity) && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinBei", "[", "z", "]"]], "\[Proportional]", RowBox[List[FractionBox["1", RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox["z"]]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["2", " ", "z"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]]]], "]"]]], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]]], SuperscriptBox["4", RowBox[List["-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "]"]], "2"], RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"], "+", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Pi]"]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"], "+", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]]]], "]"]]]]]], ")"]]]]]]]], " ", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]], SuperscriptBox["4", RowBox[List["-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["2", " ", "k"]]]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"], "+", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Pi]"]], "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"], "+", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]]]], "]"]]]]]], ")"]]]]]], "+", "\[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]"]], ")"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 bei ( z ) 1 2 π z ( 1 2 z k = 0 n - 1 2 4 - k ( - 1 z 2 ) k ( 1 2 ) 2 k + 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], Pochhammer] 2 ( 2 k + 1 ) ! ( ( - 1 ) k - z 2 cos ( π k 2 + 1 8 ( 4 2 z - π ) ) - z 2 cos ( π k 2 + 1 8 ( 4 2 z + π ) ) ) - k = 0 n 2 4 - k ( - 1 z 2 ) k ( 1 2 ) 2 k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["2", " ", "k"]]], Pochhammer] 2 ( 2 k ) ! ( - z 2 cos ( π k 2 + 1 8 ( 3 π - 4 2 z ) ) + ( - 1 ) k z 2 sin ( π k 2 + 1 8 ( π - 4 2 z ) ) ) + ) /; - π 2 < arg ( z ) π ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) n Condition Proportional KelvinBei z 1 2 1 2 z 1 2 -1 1 2 z -1 k 0 n -1 2 -1 4 -1 k -1 1 z 2 -1 k Pochhammer 1 2 2 k 1 2 2 k 1 -1 -1 k -1 z 2 1 2 -1 k 2 -1 1 8 4 2 1 2 z -1 -1 z 2 1 2 -1 k 2 -1 1 8 4 2 1 2 z -1 k 0 n 2 -1 4 -1 k -1 1 z 2 -1 k Pochhammer 1 2 2 k 2 2 k -1 -1 z 2 1 2 -1 k 2 -1 1 8 3 -1 4 2 1 2 z -1 k z 2 1 2 -1 k 2 -1 1 8 -1 4 2 1 2 z Inequality -1 2 -1 z Rule z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinBei", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]]]], "]"]]], FractionBox[RowBox[List[SuperscriptBox["4", RowBox[List["-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"], "+", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Pi]"]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"], "+", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]]]]], RowBox[List["2", " ", "z"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[SuperscriptBox["4", RowBox[List["-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["2", " ", "k"]]]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"], "+", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Pi]"]], "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"], "+", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]]]], "+", "\[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]"]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02