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 KelvinBei

 http://functions.wolfram.com/03.17.06.0034.01

 Input Form

 KelvinBei[\[Nu], z] \[Proportional] (-(1/(2 Sqrt[2 Pi] Sqrt[z]))) (Sum[((Pochhammer[1/2 - \[Nu], 2 k] Pochhammer[1/2 + \[Nu], 2 k])/(2 k)!) (I/(4 z^2))^k (E^(z/Sqrt[2]) (E^(-((I z)/Sqrt[2]) - (I Pi \[Nu])/2 - (3 Pi I)/8) + (-1)^k E^((I z)/Sqrt[2] + (I Pi \[Nu])/2 + (3 Pi I)/8)) + ((-(-1)^k) E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2 - (Pi I)/8) + E^((I z)/Sqrt[2] + (I Pi \[Nu])/2 + (Pi I)/8))/E^(z/Sqrt[2])), {k, 0, Floor[n/2]}] + (1/(2 z)) Sum[((Pochhammer[1/2 - \[Nu], 1 + 2 k] Pochhammer[1/2 + \[Nu], 1 + 2 k])/ (1 + 2 k)!) (I/(4 z^2))^k (E^(z/Sqrt[2]) (E^(-((I z)/Sqrt[2]) - (I Pi \[Nu])/2 - (Pi I)/8) + (-1)^k E^((I z)/Sqrt[2] + (I Pi \[Nu])/2 + (Pi I)/8)) + ((-1)^k E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2 - (3 Pi I)/8) - E^((I z)/Sqrt[2] + (I Pi \[Nu])/2 + (3 Pi I)/8))/E^(z/Sqrt[2])), {k, 0, Floor[(n - 1)/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", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox["z"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["2", " ", "k"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]], SuperscriptBox[RowBox[List["(", FractionBox["\[ImaginaryI]", RowBox[List["4", SuperscriptBox["z", "2"]]]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], 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[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], 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RowBox[List["2", " ", "k"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "]"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox["\[ImaginaryI]", RowBox[List["4", SuperscriptBox["z", "2"]]]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"], "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "8"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", 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[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], 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"]]]]]], "-", 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"]]]]]], ")"]]]]]], ")"]]]]]]]], "+", "\[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 ν ( z ) - 1 2 2 π z ( k = 0 n 2 ( 1 2 - ν ) 2 k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]], RowBox[List["2", " ", "k"]]], Pochhammer] ( ν + 1 2 ) 2 k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["2", " ", "k"]]], Pochhammer] ( 2 k ) ! ( 4 z 2 ) k ( - z 2 ( - ( - 1 ) k - z 2 + 3 π ν 2 - π 8 + z 2 + π ν 2 + π 8 ) + z 2 ( - z 2 - π ν 2 - 3 π 8 + ( - 1 ) k z 2 + π ν 2 + 3 π 8 ) ) + 1 2 z k = 0 n - 1 2 ( 1 2 - ν ) 2 k + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], Pochhammer] ( ν + 1 2 ) 2 k + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], Pochhammer] ( 2 k + 1 ) ! ( 4 z 2 ) k ( z 2 ( - z 2 - π ν 2 - π 8 + ( - 1 ) k z 2 + π ν 2 + π 8 ) + - z 2 ( ( - 1 ) k - z 2 + 3 π ν 2 - 3 π 8 - z 2 + π ν 2 + 3 π 8 ) ) + ) /; - π 2 < arg ( z ) π ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) n Condition Proportional KelvinBei ν z -1 1 2 2 1 2 z 1 2 -1 k 0 n 2 -1 Pochhammer 1 2 -1 ν 2 k Pochhammer ν 1 2 2 k 2 k -1 4 z 2 -1 k -1 z 2 1 2 -1 -1 -1 k -1 z 2 1 2 -1 3 ν 2 -1 -1 8 -1 z 2 1 2 -1 ν 2 -1 8 -1 z 2 1 2 -1 -1 z 2 1 2 -1 -1 ν 2 -1 -1 3 8 -1 -1 k z 2 1 2 -1 ν 2 -1 3 8 -1 1 2 z -1 k 0 n -1 2 -1 Pochhammer 1 2 -1 ν 2 k 1 Pochhammer ν 1 2 2 k 1 2 k 1 -1 4 z 2 -1 k z 2 1 2 -1 -1 z 2 1 2 -1 -1 ν 2 -1 -1 8 -1 -1 k z 2 1 2 -1 ν 2 -1 8 -1 -1 z 2 1 2 -1 -1 k -1 z 2 1 2 -1 3 ν 2 -1 -1 3 8 -1 -1 z 2 1 2 -1 ν 2 -1 3 8 -1 Inequality -1 2 -1 z Rule z n [/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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["2", " ", "k"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["\[ImaginaryI]", RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", 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RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["\[ImaginaryI]", RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"], "-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "8"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"], "+", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02