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 KelvinBei

 http://functions.wolfram.com/03.17.06.0046.01

 Input Form

 KelvinBei[\[Nu], z] \[Proportional] (-((I E^(I Pi \[Nu]))/(Sqrt[2 Pi] Sqrt[-z]))) (Sum[((Pochhammer[1/2 - \[Nu], 2 k] Pochhammer[1/2 + \[Nu], 2 k])/(2 k)!) (1/(4 z^2))^k (((-I) Sin[(1/8) (Pi (1 - 4 \[Nu]) + 4 Sqrt[2] z) + (k Pi)/2])/E^(z/Sqrt[2]) + E^(z/Sqrt[2] + Pi I \[Nu]) Sin[(1/8) (Pi (1 - 4 \[Nu]) - 4 Sqrt[2] z) + (k Pi)/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)!) (1/(4 z^2))^k ((I Cos[(1/8) ((-Pi) (1 + 4 \[Nu]) + 4 Sqrt[2] z) + (k Pi)/2])/ E^(z/Sqrt[2]) + E^(z/Sqrt[2] + Pi I \[Nu]) (-1)^k Cos[(1/8) (Pi (1 + 4 \[Nu]) + 4 Sqrt[2] z) + (k Pi)/2]), {k, 0, Floor[(n - 1)/2]}] + \[Ellipsis]) /; (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[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], SqrtBox[RowBox[List["-", "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["1", RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], ")"]], "k"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "+", FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["z", SqrtBox["2"]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Nu]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "+", FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"]]], "]"]]]]]], ")"]]]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", " ", "z"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List["1", "+", 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["1", RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], ")"]], "k"], RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[Pi]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "+", FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["z", SqrtBox["2"]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Nu]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "+", FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"]]], "]"]]]]]], ")"]]]]]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["-", "\[Infinity]"]]]], ")"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 bei ν ( z ) - π ν 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 ) ! ( 1 4 z 2 ) k ( z 2 + π ν sin ( π k 2 + 1 8 ( π ( 1 - 4 ν ) - 4 2 z ) ) - - z 2 sin ( π k 2 + 1 8 ( 4 2 z + π ( 1 - 4 ν ) ) ) ) + 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 ) ! ( 1 4 z 2 ) k ( - z 2 cos ( π k 2 + 1 8 ( 4 2 z - π ( 4 ν + 1 ) ) ) + z 2 + π ν ( - 1 ) k cos ( π k 2 + 1 8 ( 4 2 z + π ( 4 ν + 1 ) ) ) ) + ) /; ( z "\[Rule]" - ) n Condition Proportional KelvinBei ν z -1 ν 2 1 2 -1 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 1 4 z 2 -1 k z 2 1 2 -1 ν k 2 -1 1 8 1 -1 4 ν -1 4 2 1 2 z -1 -1 z 2 1 2 -1 k 2 -1 1 8 4 2 1 2 z 1 -1 4 ν 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 1 4 z 2 -1 k -1 z 2 1 2 -1 k 2 -1 1 8 4 2 1 2 z -1 4 ν 1 z 2 1 2 -1 ν -1 k k 2 -1 1 8 4 2 1 2 z 4 ν 1 Rule z -1 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["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], ")"]], " ", RowBox[List["(", 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["1", RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "+", FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["z", SqrtBox["2"]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Nu]"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "+", FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]]]], "+", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["1", RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[Pi]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "+", FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["z", SqrtBox["2"]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "+", FractionBox[RowBox[List["k", " ", "\[Pi]"]], "2"]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]]]]], RowBox[List["2", " ", "z"]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox[RowBox[List["-", "z"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["-", "\[Infinity]"]]]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02