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 KelvinBer

 http://functions.wolfram.com/03.18.03.0041.01

 Input Form

 KelvinBer[\[Nu], z] == ((z^\[Nu] Gamma[-(2/3)] Sign[\[Nu]])/ (E^((3/4) I Pi \[Nu]) ((-1)^(1/4) z)^\[Nu] (2 Gamma[1 - Abs[\[Nu]]]))) ((2^Abs[\[Nu]]/(((-1)^(1/4) z)^Abs[\[Nu]] (3 3^(2/3)))) Sum[(((I z^2)^k (-(2/3) - k + Abs[\[Nu]])!)/ (4^k (k! (-(2/3) - 2 k + Abs[\[Nu]])! Pochhammer[2/3, k] Pochhammer[1 - Abs[\[Nu]], k]))) ((-I^((-(2/3) + Abs[\[Nu]]) (1 + Sign[\[Nu]]))) (3 AiryAiPrime[(-(1/2)) 3^(2/3) ((1 + I) z)^(2/3)] + Sqrt[3] Sign[\[Nu]] AiryBiPrime[(-(1/2)) 3^(2/3) ((1 + I) z)^(2/3)]) - (-1)^k E^((3 I Pi \[Nu])/2) (3 AiryAiPrime[(1/2) 3^(2/3) ((1 + I) z)^(2/3)] + Sqrt[3] Sign[\[Nu]] AiryBiPrime[(1/2) 3^(2/3) ((1 + I) z)^(2/3)])), {k, 0, -(2/3) + Abs[\[Nu]]}] + 2^(-(7/3) + Abs[\[Nu]]) 3^(1/6) ((-1)^(1/4) z)^(4/3 - Abs[\[Nu]]) Sum[(((I z^2)^k (-(5/3) - k + Abs[\[Nu]])!)/ (4^k (k! (-(5/3) - 2 k + Abs[\[Nu]])! Pochhammer[5/3, k] Pochhammer[1 - Abs[\[Nu]], k]))) (I^((-(2/3) + Abs[\[Nu]]) (1 + Sign[\[Nu]])) (Sqrt[3] AiryAi[(-(1/2)) 3^(2/3) ((1 + I) z)^(2/3)] + Sign[\[Nu]] AiryBi[(-(1/2)) 3^(2/3) ((1 + I) z)^(2/3)]) + (-1)^k E^((3 I Pi \[Nu])/2) (Sqrt[3] AiryAi[(1/2) 3^(2/3) ((1 + I) z)^(2/3)] + Sign[\[Nu]] AiryBi[(1/2) 3^(2/3) ((1 + I) z)^(2/3)])), {k, 0, -(5/3) + Abs[\[Nu]]}]) /; Element[Abs[\[Nu]] - 2/3, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[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["2", "3"]]], "]"]], " ", RowBox[List["Sign", "[", "\[Nu]", "]"]]]], 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["Abs", "[", "\[Nu]", "]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", 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RowBox[List[RowBox[List[SqrtBox["3"], " ", RowBox[List["AiryAi", "[", 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[RowBox[List["Sign", "[", "\[Nu]", "]"]], RowBox[List["AiryBi", "[", 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["2", "3"]]], ",", "Integers"]], "]"]]]]]]

 MathML Form

 ber ν ( z ) - 3 π ν 4 z ν ( - 1 4 z ) - ν Γ ( - 2 3 ) sgn ( ν ) 2 Γ ( 1 - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ) ( 2 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 7 3 3 6 ( - 1 4 z ) 4 3 - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" k = 0 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 5 3 4 - k ( z 2 ) k ( - k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 5 3 ) ! k ! ( - 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 5 3 ) ! ( 5 3 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["5", "3"], ")"]], "k"], Pochhammer] ( 1 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[LeftBracketingBar]", "\[Nu]", "\[RightBracketingBar]"]]]], ")"]], "k"], Pochhammer] ( ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 3 ) ( sgn ( ν ) + 1 ) ( 3 Ai ( - 1 2 3 2 / 3 ( ( 1 + ) z ) 2 / 3 ) + Bi ( - 1 2 3 2 / 3 ( ( 1 + ) z ) 2 / 3 ) sgn ( ν ) ) + ( - 1 ) k 3 π ν 2 ( 3 Ai ( 1 2 3 2 / 3 ( ( 1 + ) z ) 2 / 3 ) + Bi ( 1 2 3 2 / 3 ( ( 1 + ) z ) 2 / 3 ) sgn ( ν ) ) ) + 2 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" ( - 1 4 z ) - "\[LeftBracketingBar]" ν "\[RightBracketingBar]" 3 3 2 / 3 k = 0 "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 3 4 - k ( z 2 ) k ( - k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 3 ) ! k ! ( - 2 k + "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 3 ) ! ( 2 3 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], "k"], Pochhammer] ( 1 - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[LeftBracketingBar]", "\[Nu]", "\[RightBracketingBar]"]]]], ")"]], "k"], Pochhammer] ( - ( "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 3 ) ( sgn ( ν ) + 1 ) ( 3 Ai ( - 1 2 3 2 / 3 ( ( 1 + ) z ) 2 / 3 ) + 3 Bi ( - 1 2 3 2 / 3 ( ( 1 + ) z ) 2 / 3 ) sgn ( ν ) ) - ( - 1 ) k 3 π ν 2 ( 3 Ai ( 1 2 3 2 / 3 ( ( 1 + ) z ) 2 / 3 ) + 3 Bi ( 1 2 3 2 / 3 ( ( 1 + ) z ) 2 / 3 ) sgn ( ν ) ) ) ) /; "\[LeftBracketingBar]" ν "\[RightBracketingBar]" - 2 3 TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition KelvinBer ν z -1 3 ν 4 -1 z ν -1 1 4 z -1 ν Gamma -1 2 3 Sign ν 2 Gamma 1 -1 ν -1 2 ν -1 7 3 3 1 6 -1 1 4 z 4 3 -1 ν k 0 ν -1 5 3 4 -1 k z 2 k -1 k ν -1 5 3 k -2 k ν -1 5 3 Pochhammer 5 3 k Pochhammer 1 -1 ν k -1 ν -1 2 3 Sign ν 1 3 1 2 AiryAi -1 1 2 3 2 3 1 z 2 3 AiryBi -1 1 2 3 2 3 1 z 2 3 Sign ν -1 k 3 ν 2 -1 3 1 2 AiryAi 1 2 3 2 3 1 z 2 3 AiryBi 1 2 3 2 3 1 z 2 3 Sign ν 2 ν -1 1 4 z -1 ν 3 3 2 3 -1 k 0 ν -1 2 3 4 -1 k z 2 k -1 k ν -1 2 3 k -2 k ν -1 2 3 Pochhammer 2 3 k Pochhammer 1 -1 ν k -1 -1 ν -1 2 3 Sign ν 1 3 AiryAiPrime -1 1 2 3 2 3 1 z 2 3 3 1 2 AiryBiPrime -1 1 2 3 2 3 1 z 2 3 Sign ν -1 -1 k 3 ν 2 -1 3 AiryAiPrime 1 2 3 2 3 1 z 2 3 3 1 2 AiryBiPrime 1 2 3 2 3 1 z 2 3 Sign ν ν -1 2 3 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[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["2", "3"]]], "]"]], " ", RowBox[List["Sign", "[", "\[Nu]", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", 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["2", "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["2", "3"]]], "-", "k", "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], "!"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ImaginaryI]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["2", "3"]]], "+", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["Sign", "[", "\[Nu]", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02