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 KelvinBer

 http://functions.wolfram.com/03.18.06.0051.01

 Input Form

 KelvinBer[\[Nu], z] \[Proportional] (((-1)^(1/4) z^\[Nu])/(E^((I Pi \[Nu])/4) (2 Sqrt[2 Pi]))) ((E^(z/Sqrt[2]) (E^((I z)/Sqrt[2] + (3 I Pi \[Nu])/2) ((-1)^(3/4) z)^(-(1/2) - \[Nu]) (Sum[((Pochhammer[1/2 - \[Nu], 2 k] Pochhammer[1/2 + \[Nu], 2 k])/ (2^(2 k) (2 k)!)) (-(I/z^2))^k, {k, 0, Floor[n/2]}] + O[1/z^(2 Floor[n/2] + 2)]) - (((-(-1)^(1/4)) z)^(-(1/2) - \[Nu]) ((((-1)^(3/4) Sqrt[(-I) z^2])/z) Cos[Pi \[Nu]] + Sin[Pi \[Nu]]) (Sum[((Pochhammer[1/2 - \[Nu], 2 k] Pochhammer[1/2 + \[Nu], 2 k])/ (2^(2 k) (2 k)!)) (I/z^2)^k, {k, 0, Floor[n/2]}] + O[1/z^(2 Floor[n/2] + 2)]))/E^((I z)/Sqrt[2])) + (E^((I z)/Sqrt[2]) ((-(-1)^(1/4)) z)^(-(1/2) - \[Nu]) (Sum[((Pochhammer[1/2 - \[Nu], 2 k] Pochhammer[1/2 + \[Nu], 2 k])/ (2^(2 k) (2 k)!)) (I/z^2)^k, {k, 0, Floor[n/2]}] + O[1/z^(2 Floor[n/2] + 2)]) + E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2) ((-1)^(3/4) z)^(-(1/2) - \[Nu]) ((((-1)^(1/4) Sqrt[I z^2])/z) Cos[Pi \[Nu]] - Sin[Pi \[Nu]]) (Sum[((Pochhammer[1/2 - \[Nu], 2 k] Pochhammer[1/2 + \[Nu], 2 k])/ (2^(2 k) (2 k)!)) (-(I/z^2))^k, {k, 0, Floor[n/2]}] + O[1/z^(2 Floor[n/2] + 2)]))/E^(z/Sqrt[2])) + ((-1)^(3/4)/z) (E^(z/Sqrt[2]) ((-E^((I z)/Sqrt[2] + (3 I Pi \[Nu])/2)) ((-1)^(3/4) z)^(-(1/2) - \[Nu]) (Sum[((2^(-1 - 2 k) Pochhammer[1/2 - \[Nu], 1 + 2 k] Pochhammer[1/2 + \[Nu], 1 + 2 k])/(1 + 2 k)!) (-(I/z^2))^k, {k, 0, Floor[(n - 1)/2]}] + O[1/z^(2 Floor[(n - 1)/2] + 2)]) + (I ((-(-1)^(1/4)) z)^(-(1/2) - \[Nu]) ((((-1)^(3/4) Sqrt[(-I) z^2])/z) Cos[Pi \[Nu]] + Sin[Pi \[Nu]]) (Sum[((2^(-1 - 2 k) Pochhammer[1/2 - \[Nu], 1 + 2 k] Pochhammer[ 1/2 + \[Nu], 1 + 2 k])/(1 + 2 k)!) (I/z^2)^k, {k, 0, Floor[(n - 1)/2]}] + O[1/z^(2 Floor[(n - 1)/2] + 2)]))/ E^((I z)/Sqrt[2])) + (I E^((I z)/Sqrt[2]) ((-(-1)^(1/4)) z)^ (-(1/2) - \[Nu]) (Sum[((2^(-1 - 2 k) Pochhammer[1/2 - \[Nu], 1 + 2 k] Pochhammer[1/2 + \[Nu], 1 + 2 k])/(1 + 2 k)!) (I/z^2)^k, {k, 0, Floor[(n - 1)/2]}] + O[1/z^(2 Floor[(n - 1)/2] + 2)]) + E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2) ((-1)^(3/4) z)^ (-(1/2) - \[Nu]) ((((-1)^(1/4) Sqrt[I z^2])/z) Cos[Pi \[Nu]] - Sin[Pi \[Nu]]) (Sum[((2^(-1 - 2 k) Pochhammer[1/2 - \[Nu], 1 + 2 k] Pochhammer[1/2 + \[Nu], 1 + 2 k])/(1 + 2 k)!) (-(I/z^2))^k, {k, 0, Floor[(n - 1)/2]}] + O[1/z^(2 Floor[(n - 1)/2] + 2)]))/ E^(z/Sqrt[2]))) /; (Abs[z] -> Infinity) && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "4"]]]], SuperscriptBox["z", "\[Nu]"], " "]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], RowBox[List["(", " ", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]], "+", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], 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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["\[ImaginaryI]", SuperscriptBox["z", "2"]], ")"]], "k"]]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List[RowBox[List["2", RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]]], "+", "2"]]]], "]"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]]]], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[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["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]", SuperscriptBox["z", "2"]], ")"]], "k"]]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List[RowBox[List["2", RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]]], "+", "2"]]]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]]], "+", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["z", "2"]]]]]], "z"], RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "-", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[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["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["(", RowBox[List["-", FractionBox["\[ImaginaryI]", SuperscriptBox["z", "2"]]]], ")"]], "k"]]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", RowBox[List[RowBox[List["2", RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]]], "+", "2"]]]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 ber ν ( z ) - 1 4 - π ν 4 z ν 2 2 π ( ( - z 2 ( z 2 ( - - 1 4 z ) - ν - 1 2 ( k = 0 n 2 2 - 2 k ( 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 ) ! ( z 2 ) k + O ( 1 z 2 n 2 + 2 ) ) + 3 π ν 2 - z 2 ( ( - 1 ) 3 / 4 z ) - ν - 1 2 ( - 1 4 z 2 z cos ( π ν ) - sin ( π ν ) ) ( k = 0 n 2 2 - 2 k ( 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 ) ! ( - z 2 ) k + O ( 1 z 2 n 2 + 2 ) ) ) + z 2 ( z 2 + 3 π ν 2 ( ( - 1 ) 3 / 4 z ) - ν - 1 2 ( k = 0 n 2 2 - 2 k ( 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 ) ! ( - z 2 ) k + O ( 1 z 2 n 2 + 2 ) ) - - z 2 ( - - 1 4 z ) - ν - 1 2 ( ( - 1 ) 3 / 4 - z 2 z cos ( π ν ) + sin ( π ν ) ) ( k = 0 n 2 2 - 2 k ( 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 ) ! ( z 2 ) k + O ( 1 z 2 n 2 + 2 ) ) ) ) + ( - 1 ) 3 / 4 z ( - z 2 ( z 2 ( - - 1 4 z ) - ν - 1 2 ( k = 0 n - 1 2 ( 2 - 2 k - 1 ( 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 ) ! ( z 2 ) k + O ( 1 z 2 n - 1 2 + 2 ) ) + 3 π ν 2 - z 2 ( ( - 1 ) 3 / 4 z ) - ν - 1 2 ( - 1 4 z 2 z cos ( π ν ) - sin ( π ν ) ) ( k = 0 n - 1 2 ( 2 - 2 k - 1 ( 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 ) ! ( - z 2 ) k + O ( 1 z 2 n - 1 2 + 2 ) ) ) + z 2 ( - z 2 ( - - 1 4 z ) - ν - 1 2 ( ( - 1 ) 3 / 4 - z 2 z cos ( π ν ) + sin ( π ν ) ) ( k = 0 n - 1 2 ( 2 - 2 k - 1 ( 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 ) ! ( z 2 ) k + O ( 1 z 2 n - 1 2 + 2 ) ) - z 2 + 3 π ν 2 ( ( - 1 ) 3 / 4 z ) - ν - 1 2 ( k = 0 n - 1 2 ( 2 - 2 k - 1 ( 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 ) ! ( - z 2 ) k + O ( 1 z 2 n - 1 2 + 2 ) ) ) ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) n Condition Proportional KelvinBer ν z -1 1 4 -1 ν 4 -1 z ν 2 2 1 2 -1 -1 z 2 1 2 -1 z 2 1 2 -1 -1 -1 1 4 z -1 ν -1 1 2 k 0 n 2 -1 2 -2 k Pochhammer 1 2 -1 ν 2 k Pochhammer ν 1 2 2 k 2 k -1 z 2 -1 k O 1 z 2 n 2 -1 2 -1 3 ν 2 -1 -1 z 2 1 2 -1 -1 3 4 z -1 ν -1 1 2 -1 1 4 z 2 1 2 z -1 ν -1 ν k 0 n 2 -1 2 -2 k Pochhammer 1 2 -1 ν 2 k Pochhammer ν 1 2 2 k 2 k -1 -1 z 2 -1 k O 1 z 2 n 2 -1 2 -1 z 2 1 2 -1 z 2 1 2 -1 3 ν 2 -1 -1 3 4 z -1 ν -1 1 2 k 0 n 2 -1 2 -2 k Pochhammer 1 2 -1 ν 2 k Pochhammer ν 1 2 2 k 2 k -1 -1 z 2 -1 k O 1 z 2 n 2 -1 2 -1 -1 -1 z 2 1 2 -1 -1 -1 1 4 z -1 ν -1 1 2 -1 3 4 -1 z 2 1 2 z -1 ν ν k 0 n 2 -1 2 -2 k Pochhammer 1 2 -1 ν 2 k Pochhammer ν 1 2 2 k 2 k -1 z 2 -1 k O 1 z 2 n 2 -1 2 -1 -1 3 4 z -1 -1 z 2 1 2 -1 z 2 1 2 -1 -1 -1 1 4 z -1 ν -1 1 2 k 0 n -1 2 -1 2 -2 k -1 Pochhammer 1 2 -1 ν 2 k 1 Pochhammer ν 1 2 2 k 1 2 k 1 -1 z 2 -1 k O 1 z 2 n -1 2 -1 2 -1 3 ν 2 -1 -1 z 2 1 2 -1 -1 3 4 z -1 ν -1 1 2 -1 1 4 z 2 1 2 z -1 ν -1 ν k 0 n -1 2 -1 2 -2 k -1 Pochhammer 1 2 -1 ν 2 k 1 Pochhammer ν 1 2 2 k 1 2 k 1 -1 -1 z 2 -1 k O 1 z 2 n -1 2 -1 2 -1 z 2 1 2 -1 -1 z 2 1 2 -1 -1 -1 1 4 z -1 ν -1 1 2 -1 3 4 -1 z 2 1 2 z -1 ν ν k 0 n -1 2 -1 2 -2 k -1 Pochhammer 1 2 -1 ν 2 k 1 Pochhammer ν 1 2 2 k 1 2 k 1 -1 z 2 -1 k O 1 z 2 n -1 2 -1 2 -1 -1 z 2 1 2 -1 3 ν 2 -1 -1 3 4 z -1 ν -1 1 2 k 0 n -1 2 -1 2 -2 k -1 Pochhammer 1 2 -1 ν 2 k 1 Pochhammer ν 1 2 2 k 1 2 k 1 -1 -1 z 2 -1 k O 1 z 2 n -1 2 -1 2 -1 Rule z n [/itex]

 Rule Form

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

 2007-05-02