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 KelvinBei

 http://functions.wolfram.com/03.17.03.0039.01

 Input Form

 KelvinBei[\[Nu], z] == ((-1)^(3/8)/(E^(I Pi \[Nu]) (Sqrt[2 Pi] Sqrt[z]))) (Sum[(((Abs[\[Nu]] + 2 k + 1/2)! (2 (-1)^(1/4) z)^(-2 k - 1))/ ((2 k + 1)! (Abs[\[Nu]] - 2 k - 3/2)!)) (E^((1/4) I Pi (1 + 4 \[Nu])) Cos[(1/2) Pi (1/2 - \[Nu]) - z/(-1)^4^(-1)] + (-1)^k Cos[(1/2) Pi (\[Nu] - 1/2) - (-1)^(1/4) z]), {k, 0, Floor[(1/4) (2 Abs[\[Nu]] - 3)]}] + Sum[((Abs[\[Nu]] + 2 k - 1/2)!/((2 (-1)^(1/4) z)^(2 k) ((2 k)! (Abs[\[Nu]] - 2 k - 1/2)!))) ((-1)^(3/4) E^(I Pi \[Nu]) Sin[(1/2) Pi (1/2 - \[Nu]) - z/(-1)^4^(-1)] - (-1)^k Sin[(1/2) Pi (\[Nu] - 1/2) - (-1)^(1/4) z]), {k, 0, Floor[(1/4) (2 Abs[\[Nu]] - 1)]}]) /; Element[\[Nu] - 1/2, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "8"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", "\[Nu]"]]]]], RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox["z"]]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "3"]], ")"]]]], "]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", RowBox[List["2", "k"]], "+", FractionBox["1", "2"]]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", "1"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", RowBox[List["2", "k"]], "-", FractionBox["3", "2"]]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "/", "4"]]], " ", "z"]]]], "]"]]]], " ", "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], ")"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]]]], ")"]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "-", "1"]], ")"]]]], "]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "+", RowBox[List["2", "k"]], "-", FractionBox["1", "2"]]], ")"]], "!"]], SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", RowBox[List["2", "k"]], "-", FractionBox["1", "2"]]], ")"]], "!"]], " "]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]], ")"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "/", "4"]]], " ", "z"]]]], "]"]]]], " ", "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], ")"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]]]], ")"]]]]]]]], " ", ")"]]]]]], "/;", RowBox[List[RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]], "\[Element]", "Integers"]]]]]]

 MathML Form

 bei ν ( z ) ( - 1 ) 3 / 8 - π ν 2 π z ( k = 0 1 4 ( 2 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 3 ) ( 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" + 1 2 ) ! ( 2 - 1 4 z ) - 2 k - 1 ( 2 k + 1 ) ! ( - 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 3 2 ) ! ( 1 4 π ( 4 ν + 1 ) cos ( 1 2 π ( 1 2 - ν ) - 1 - 1 4 z ) + ( - 1 ) k cos ( 1 2 π ( ν - 1 2 ) - - 1 4 z ) ) + k = 0 1 4 ( 2 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 ) ( 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 2 ) ! ( 2 - 1 4 z ) - 2 k ( 2 k ) ! ( - 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 2 ) ! ( ( - 1 ) 3 / 4 π ν sin ( 1 2 π ( 1 2 - ν ) - 1 - 1 4 z ) - ( - 1 ) k sin ( 1 2 π ( ν - 1 2 ) - - 1 4 z ) ) ) /; ν - 1 2 TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition KelvinBei ν z -1 3 8 -1 ν 2 1 2 z 1 2 -1 k 0 1 4 2 ν -3 2 k ν 1 2 2 -1 1 4 z -2 k -1 2 k 1 -2 k ν -1 3 2 -1 1 4 4 ν 1 1 2 1 2 -1 ν -1 1 -1 1 4 -1 z -1 k 1 2 ν -1 1 2 -1 -1 1 4 z k 0 1 4 2 ν -1 2 k ν -1 1 2 2 -1 1 4 z -2 k 2 k -2 k ν -1 1 2 -1 -1 3 4 ν 1 2 1 2 -1 ν -1 1 -1 1 4 -1 z -1 -1 k 1 2 ν -1 1 2 -1 -1 1 4 z ν -1 1 2 [/itex]

 Rule Form

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

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

 2007-05-02