html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 KelvinBer

 http://functions.wolfram.com/03.14.06.0036.01

 Input Form

 KelvinBer[z] \[Proportional] ((-1)^(1/4)/(2 Sqrt[2 Pi])) (E^(z/Sqrt[2]) ((E^((I z)/Sqrt[2])/Sqrt[(-1)^(3/4) z]) (Sum[(Pochhammer[1/2, 2 k]^2/(2 k)!) (-(I/(4 z^2)))^k, {k, 0, Floor[n/2]}] + O[1/z^(2 Floor[n/2] + 2)]) + (1/(E^((I z)/Sqrt[2]) Sqrt[(-1)^(1/4) z])) (Sum[(Pochhammer[1/2, 2 k]^2/(2 k)!) (I/(4 z^2))^k, {k, 0, Floor[n/2]}] + O[1/z^(2 Floor[n/2] + 2)])) + ((E^((I z)/Sqrt[2])/Sqrt[(-(-1)^(1/4)) z]) (Sum[(Pochhammer[1/2, 2 k]^2/(2 k)!) (I/(4 z^2))^k, {k, 0, Floor[n/2]}] + O[1/z^(2 Floor[n/2] + 2)]) + (1/(E^((I z)/Sqrt[2]) Sqrt[(-(-1)^(3/4)) z])) (Sum[(Pochhammer[1/2, 2 k]^2/(2 k)!) (-(I/(4 z^2)))^k, {k, 0, Floor[n/2]}] + O[1/z^(2 Floor[n/2] + 2)]))/E^(z/Sqrt[2]) + ((-1)^(3/4)/(2 z)) ((-E^(z/Sqrt[2])) ((E^((I z)/Sqrt[2])/Sqrt[(-1)^(3/4) z]) (Sum[(Pochhammer[1/2, 1 + 2 k]^2/(1 + 2 k)!) (-(I/(4 z^2)))^k, {k, 0, Floor[(n - 1)/2]}] + O[1/z^(2 Floor[(n - 1)/2] + 2)]) + (I/(E^((I z)/Sqrt[2]) Sqrt[(-1)^(1/4) z])) (Sum[(Pochhammer[1/2, 1 + 2 k]^2/(1 + 2 k)!) (I/(4 z^2))^k, {k, 0, Floor[(n - 1)/2]}] + O[1/z^(2 Floor[(n - 1)/2] + 2)])) + (((I E^((I z)/Sqrt[2]))/Sqrt[(-(-1)^(1/4)) z]) (Sum[(Pochhammer[1/2, 1 + 2 k]^2/(1 + 2 k)!) (I/(4 z^2))^k, {k, 0, Floor[(n - 1)/2]}] + O[1/z^(2 Floor[(n - 1)/2] + 2)]) + (1/(E^((I z)/Sqrt[2]) Sqrt[(-(-1)^(3/4)) z])) (Sum[(Pochhammer[1/2, 1 + 2 k]^2/(1 + 2 k)!) (-(I/(4 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", "[", "z", "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " "]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]], SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", "z"]]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["2", " ", "k"]]]], 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 MathML Form

 ber ( z ) - 1 4 2 2 π ( z 2 ( z 2 ( - 1 ) 3 / 4 z ( k = 0 n 2 ( 1 2 ) 2 k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["2", " ", "k"]]], Pochhammer] 2 ( 2 k ) ! ( - 4 z 2 ) k + O ( 1 z 2 n 2 + 2 ) ) + - z 2 - 1 4 z ( k = 0 n 2 ( 1 2 ) 2 k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["2", " ", "k"]]], Pochhammer] 2 ( 2 k ) ! ( 4 z 2 ) k + O ( 1 z 2 n 2 + 2 ) ) ) + - z 2 ( - z 2 - ( - 1 ) 3 / 4 z ( k = 0 n 2 ( 1 2 ) 2 k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["2", " ", "k"]]], Pochhammer] 2 ( 2 k ) ! ( - 4 z 2 ) k + O ( 1 z 2 n 2 + 2 ) ) + z 2 - - 1 4 z ( k = 0 n 2 ( 1 2 ) 2 k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["2", " ", "k"]]], Pochhammer] 2 ( 2 k ) ! ( 4 z 2 ) k + O ( 1 z 2 n 2 + 2 ) ) ) + ( - 1 ) 3 / 4 2 z ( - z 2 ( z 2 - - 1 4 z ( k = 0 n - 1 2 ( 1 2 ) 2 k + 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], Pochhammer] 2 ( 2 k + 1 ) ! ( 4 z 2 ) k + O ( 1 z 2 n - 1 2 + 2 ) ) + - z 2 - ( - 1 ) 3 / 4 z ( k = 0 n - 1 2 ( 1 2 ) 2 k + 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], Pochhammer] 2 ( 2 k + 1 ) ! ( - 4 z 2 ) k + O ( 1 z 2 n - 1 2 + 2 ) ) ) - z 2 ( z 2 ( - 1 ) 3 / 4 z ( k = 0 n - 1 2 ( 1 2 ) 2 k + 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], Pochhammer] 2 ( 2 k + 1 ) ! ( - 4 z 2 ) k + O ( 1 z 2 n - 1 2 + 2 ) ) + - z 2 - 1 4 z ( k = 0 n - 1 2 ( 1 2 ) 2 k + 1 TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], Pochhammer] 2 ( 2 k + 1 ) ! ( 4 z 2 ) k + O ( 1 z 2 n - 1 2 + 2 ) ) ) ) ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) n Condition Proportional KelvinBer z -1 1 4 2 2 1 2 -1 z 2 1 2 -1 z 2 1 2 -1 -1 3 4 z 1 2 -1 k 0 n 2 -1 Pochhammer 1 2 2 k 2 2 k -1 -1 4 z 2 -1 k O 1 z 2 n 2 -1 2 -1 -1 z 2 1 2 -1 -1 1 4 z 1 2 -1 k 0 n 2 -1 Pochhammer 1 2 2 k 2 2 k -1 4 z 2 -1 k O 1 z 2 n 2 -1 2 -1 -1 z 2 1 2 -1 -1 z 2 1 2 -1 -1 -1 3 4 z 1 2 -1 k 0 n 2 -1 Pochhammer 1 2 2 k 2 2 k -1 -1 4 z 2 -1 k O 1 z 2 n 2 -1 2 -1 z 2 1 2 -1 -1 -1 1 4 z 1 2 -1 k 0 n 2 -1 Pochhammer 1 2 2 k 2 2 k -1 4 z 2 -1 k O 1 z 2 n 2 -1 2 -1 -1 3 4 2 z -1 -1 z 2 1 2 -1 z 2 1 2 -1 -1 -1 1 4 z 1 2 -1 k 0 n -1 2 -1 Pochhammer 1 2 2 k 1 2 2 k 1 -1 4 z 2 -1 k O 1 z 2 n -1 2 -1 2 -1 -1 z 2 1 2 -1 -1 -1 3 4 z 1 2 -1 k 0 n -1 2 -1 Pochhammer 1 2 2 k 1 2 2 k 1 -1 -1 4 z 2 -1 k O 1 z 2 n -1 2 -1 2 -1 -1 z 2 1 2 -1 z 2 1 2 -1 -1 3 4 z 1 2 -1 k 0 n -1 2 -1 Pochhammer 1 2 2 k 1 2 2 k 1 -1 -1 4 z 2 -1 k O 1 z 2 n -1 2 -1 2 -1 -1 z 2 1 2 -1 -1 1 4 z 1 2 -1 k 0 n -1 2 -1 Pochhammer 1 2 2 k 1 2 2 k 1 -1 4 z 2 -1 k O 1 z 2 n -1 2 -1 2 -1 Rule z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinBer", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], SqrtBox["2"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["2", " ", "k"]]]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[ImaginaryI]", RowBox[List["4", " ", SuperscriptBox["z", "2"]]]]]], ")"]], "k"]]], 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02