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 KelvinBei

 http://functions.wolfram.com/03.17.03.0040.01

 Input Form

 KelvinBei[\[Nu], z] == (-((I z^\[Nu] Gamma[-(1/3)])/(E^((3/4) I Pi \[Nu]) ((-1)^(1/4) z)^\[Nu] (2 Gamma[1 - Abs[\[Nu]]])))) ((2^(-1 + Abs[\[Nu]])/(((-1)^(1/4) z)^Abs[\[Nu]] 3^(5/6))) Sum[(((I z^2)^k (-(1/3) - k + Abs[\[Nu]])!)/ (4^k (k! (-(1/3) - 2 k + Abs[\[Nu]])! Pochhammer[1/3, k] Pochhammer[1 - Abs[\[Nu]], k]))) (I^((-(1/3) + Abs[\[Nu]]) (1 + Sign[\[Nu]])) Sign[\[Nu]] (-AiryBi[(-(1/2)) 3^(2/3) ((1 + I) z)^(2/3)] + Sqrt[3] Sign[\[Nu]] AiryAi[(-(1/2)) 3^(2/3) ((1 + I) z)^(2/3)]) + (-1)^k E^((3 I Pi \[Nu])/2) (-AiryBi[(1/2) 3^(2/3) ((1 + I) z)^(2/3)] + Sqrt[3] Sign[\[Nu]] AiryAi[(1/2) 3^(2/3) ((1 + I) z)^(2/3)])), {k, 0, -(1/3) + Abs[\[Nu]]}] + ((2^(-(5/3) + Abs[\[Nu]]) ((-1)^(1/4) z)^(2/3 - Abs[\[Nu]]))/3^(2/3)) Sum[(((I z^2)^k (-(4/3) - k + Abs[\[Nu]])!)/ (4^k (k! (-(4/3) - 2 k + Abs[\[Nu]])! Pochhammer[4/3, k] Pochhammer[1 - Abs[\[Nu]], k]))) ((-I^((-(1/3) + Abs[\[Nu]]) (1 + Sign[\[Nu]]))) Sign[\[Nu]] (Sqrt[3] AiryBiPrime[(-(1/2)) 3^(2/3) ((1 + I) z)^(2/3)] - 3 Sign[\[Nu]] AiryAiPrime[(-(1/2)) 3^(2/3) ((1 + I) z)^(2/3)]) + (-1)^k E^((3 I Pi \[Nu])/2) (Sqrt[3] AiryBiPrime[(1/2) 3^(2/3) ((1 + I) z)^(2/3)] - 3 Sign[\[Nu]] AiryAiPrime[ (1/2) 3^(2/3) ((1 + I) z)^(2/3)])), {k, 0, -(4/3) + Abs[\[Nu]]}]) /; Element[Abs[\[Nu]] - 1/3, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["3", "4"]]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["1", "3"]]], "]"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "]"]]]]]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox[RowBox[List["(", 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RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ",", "k"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Sign", "[", "\[Nu]", "]"]]]], ")"]]]]], " ", RowBox[List["Sign", "[", "\[Nu]", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["AiryBi", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", "z"]], ")"]], RowBox[List["2", "/", "3"]]]]], "]"]]]], "+", RowBox[List[SqrtBox["3"], " ", RowBox[List["Sign", "[", 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RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", "z"]], ")"]], RowBox[List["2", "/", "3"]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["5", "3"]]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], RowBox[List[FractionBox["2", "3"], "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " "]], SuperscriptBox["3", RowBox[List["2", "/", "3"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["4", RowBox[List["-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], "-", "k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["4", "3"]]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["4", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ",", "k"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ImaginaryI]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Sign", "[", "\[Nu]", "]"]]]], ")"]]]]]]], " ", RowBox[List["Sign", "[", "\[Nu]", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["3"], " ", RowBox[List["AiryBiPrime", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", "z"]], ")"]], RowBox[List["2", "/", "3"]]]]], "]"]]]], "-", RowBox[List["3", " ", RowBox[List["Sign", "[", "\[Nu]", "]"]], RowBox[List["AiryAiPrime", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", "z"]], ")"]], RowBox[List["2", "/", "3"]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["3"], " ", RowBox[List["AiryBiPrime", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", "z"]], ")"]], RowBox[List["2", "/", "3"]]]]], "]"]]]], "-", RowBox[List["3", " ", RowBox[List["Sign", "[", "\[Nu]", "]"]], RowBox[List["AiryAiPrime", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", "z"]], ")"]], RowBox[List["2", "/", "3"]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]]], ")"]]]]]], "/;", RowBox[List["Element", "[", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", FractionBox["1", "3"]]], ",", "Integers"]], 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 MathML Form

 bei ν ( z ) - 1 4 ( - 3 ) π ν z ν ( - 1 4 z ) - ν Γ ( - 1 3 ) 2 Γ ( 1 - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ( 2 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 ( - 1 4 z ) - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" 3 5 / 6 k = 0 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 3 4 - k ( z 2 ) k ( - k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 3 ) ! k ! ( - 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 3 ) ! ( 1 3 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "3"], ")"]], "k"], Pochhammer] ( 1 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[LeftBracketingBar]", "\[Nu]", "\[RightBracketingBar]"]]]], ")"]], "k"], Pochhammer] ( ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 3 ) ( sgn ( ν ) + 1 ) sgn ( ν ) ( 3 sgn ( ν ) Ai ( - 1 2 3 2 / 3 ( ( 1 + ) z ) 2 / 3 ) - Bi ( - 1 2 3 2 / 3 ( ( 1 + ) z ) 2 / 3 ) ) + ( - 1 ) k 3 π ν 2 ( 3 sgn ( ν ) Ai ( 1 2 3 2 / 3 ( ( 1 + ) z ) 2 / 3 ) - Bi ( 1 2 3 2 / 3 ( ( 1 + ) z ) 2 / 3 ) ) ) + 2 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 5 3 ( - 1 4 z ) 2 3 - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" 3 2 / 3 k = 0 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 4 3 4 - k ( z 2 ) k ( - k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 4 3 ) ! k ! ( - 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 4 3 ) ! ( 4 3 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["4", "3"], ")"]], "k"], Pochhammer] ( 1 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[LeftBracketingBar]", "\[Nu]", "\[RightBracketingBar]"]]]], ")"]], "k"], Pochhammer] ( - ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 3 ) ( sgn ( ν ) + 1 ) sgn ( ν ) ( 3 Bi ( - 1 2 3 2 / 3 ( ( 1 + ) z ) 2 / 3 ) - 3 sgn ( ν ) Ai ( - 1 2 3 2 / 3 ( ( 1 + ) z ) 2 / 3 ) ) + ( - 1 ) k 3 π ν 2 ( 3 Bi ( 1 2 3 2 / 3 ( ( 1 + ) z ) 2 / 3 ) - 3 sgn ( ν ) Ai ( 1 2 3 2 / 3 ( ( 1 + ) z ) 2 / 3 ) ) ) ) /; "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 1 3 TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition KelvinBei ν z -1 1 4 -3 ν z ν -1 1 4 z -1 ν Gamma -1 1 3 2 Gamma 1 -1 ν -1 2 ν -1 -1 1 4 z -1 ν 3 5 6 -1 k 0 ν -1 1 3 4 -1 k z 2 k -1 k ν -1 1 3 k -2 k ν -1 1 3 Pochhammer 1 3 k Pochhammer 1 -1 ν k -1 ν -1 1 3 Sign ν 1 Sign ν 3 1 2 Sign ν AiryAi -1 1 2 3 2 3 1 z 2 3 -1 AiryBi -1 1 2 3 2 3 1 z 2 3 -1 k 3 ν 2 -1 3 1 2 Sign ν AiryAi 1 2 3 2 3 1 z 2 3 -1 AiryBi 1 2 3 2 3 1 z 2 3 2 ν -1 5 3 -1 1 4 z 2 3 -1 ν 3 2 3 -1 k 0 ν -1 4 3 4 -1 k z 2 k -1 k ν -1 4 3 k -2 k ν -1 4 3 Pochhammer 4 3 k Pochhammer 1 -1 ν k -1 -1 ν -1 1 3 Sign ν 1 Sign ν 3 1 2 AiryBiPrime -1 1 2 3 2 3 1 z 2 3 -1 3 Sign ν AiryAiPrime -1 1 2 3 2 3 1 z 2 3 -1 k 3 ν 2 -1 3 1 2 AiryBiPrime 1 2 3 2 3 1 z 2 3 -1 3 Sign ν AiryAiPrime 1 2 3 2 3 1 z 2 3 ν -1 1 3 [/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[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["-", "3"]], ")"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["1", "3"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], ")"]], RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["4", RowBox[List["-", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], "-", "k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["1", "3"]]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Sign", "[", "\[Nu]", "]"]]]], ")"]]]]], " ", RowBox[List["Sign", "[", "\[Nu]", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["AiryBi", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", "z"]], ")"]], RowBox[List["2", "/", "3"]]]]], "]"]]]], "+", RowBox[List[SqrtBox["3"], " ", RowBox[List["Sign", "[", "\[Nu]", "]"]], " ", RowBox[List["AiryAi", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", "z"]], ")"]], RowBox[List["2", "/", "3"]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02