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/03.18.20.0007.01

 Input Form

 Derivative[1, 0][KelvinBer][-n - 1/2, z] == (-(3/4)) Pi KelvinBei[-(1/2) - n, z] + KelvinBer[-(1/2) - n, z] (Log[z] - Log[(-1)^(1/4) z]) + (((-1)^(3/8) 2^(-(1/2) - n) E^((7 I n Pi)/4) z^(-(1/2) - n))/ (Sqrt[Pi] n!)) Sum[2^(2 k) Binomial[n, 2 k] (2 n - 2 k)! I^k (((PolyGamma[k + 1/2] - PolyGamma[k - n + 1/2]) Cosh[(-1)^(1/4) z] + Cosh[(-1)^(1/4) z] CoshIntegral[2 (-1)^(1/4) z] - Sinh[(-1)^(1/4) z] SinhIntegral[2 (-1)^(1/4) z])/ E^((3/4) I (1 + 2 n) Pi) + (-1)^k ((PolyGamma[k + 1/2] - PolyGamma[k - n + 1/2]) Cos[(-1)^(1/4) z] + Cos[(-1)^(1/4) z] CosIntegral[2 (-1)^(1/4) z] + Sin[(-1)^(1/4) z] SinIntegral[2 (-1)^(1/4) z])) z^(2 k), {k, 0, Floor[n/2]}] + (((-1)^(5/8) 2^(1/2 - n) z^(1/2 - n))/ (E^((I n Pi)/4) (Sqrt[Pi] n!))) Sum[2^(2 k) Binomial[n, 2 k + 1] (2 n - 2 k - 1)! I^k (((-(PolyGamma[k + 3/2] - PolyGamma[k - n + 1/2])) Sinh[(-1)^(1/4) z] + Cosh[(-1)^(1/4) z] SinhIntegral[ 2 (-1)^(1/4) z] - Sinh[(-1)^(1/4) z] CoshIntegral[2 (-1)^(1/4) z])/ E^((3/4) I (1 + 2 n) Pi) + (-1)^k ((PolyGamma[k + 3/2] - PolyGamma[k - n + 1/2]) Sin[(-1)^(1/4) z] + Sin[(-1)^(1/4) z] CosIntegral[2 (-1)^(1/4) z] - Cos[(-1)^(1/4) z] SinIntegral[2 (-1)^(1/4) z])) z^(2 k), {k, 0, Floor[(n - 1)/2]}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0"]], "]"]], "[", "KelvinBer", "]"]], "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "-", FractionBox["1", "2"]]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["3", "4"]]], " ", "\[Pi]", " ", RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]], ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "8"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["7", " ", "\[ImaginaryI]", " ", "n", " ", "\[Pi]"]], "4"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["n", "!"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[SuperscriptBox["2", RowBox[List["2", "k"]]], RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List["2", "k"]]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "-", RowBox[List["2", "k"]]]], ")"]], "!"]], SuperscriptBox["\[ImaginaryI]", "k"], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["3", "4"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", "\[Pi]"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["1", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]], RowBox[List["Cosh", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Cosh", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]], RowBox[List["CoshIntegral", "[", RowBox[List["2", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]]]], "-", RowBox[List[RowBox[List["Sinh", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]], RowBox[List["SinhIntegral", "[", RowBox[List["2", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["1", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Cos", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]], " ", RowBox[List["CosIntegral", "[", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]]]]]], ")"]]]]]], ")"]], SuperscriptBox["z", RowBox[List["2", "k"]]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["5", "/", "8"]]], " ", SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "n", " ", "\[Pi]"]], "4"]]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "-", "n"]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["n", "!"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], "!"]], " ", SuperscriptBox["\[ImaginaryI]", "k"], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["3", "4"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["3", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]]]], RowBox[List["Sinh", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Cosh", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]], RowBox[List["SinhIntegral", "[", RowBox[List["2", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]]]], "-", RowBox[List[RowBox[List["Sinh", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]], RowBox[List["CoshIntegral", "[", RowBox[List["2", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["3", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]], " ", RowBox[List["CosIntegral", "[", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]]]], "-", RowBox[List[RowBox[List["Cos", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], "z"]], "]"]]]]]], ")"]]]]]], ")"]], SuperscriptBox["z", RowBox[List["2", "k"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 ber TagBox["ber", BesselJ] - n - 1 2 ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( z ) - 3 π 4 bei - n - 1 2 ( z ) + ( log ( z ) - log ( - 1 4 z ) ) ber - n - 1 2 ( z ) + ( - 1 ) 3 / 8 2 - n - 1 2 7 n π 4 z - n - 1 2 π n ! k = 0 n 2 2 2 k ( n 2 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List["2", " ", "k"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( 2 n - 2 k ) ! k ( 1 4 ( - 3 ) ( 2 n + 1 ) π ( cosh ( - 1 4 z ) Chi ( 2 - 1 4 z ) + cosh ( - 1 4 z ) ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) - sinh ( - 1 4 z ) Shi ( 2 - 1 4 z ) ) + ( - 1 ) k ( cos ( - 1 4 z ) Ci ( 2 - 1 4 z ) + cos ( - 1 4 z ) ( ψ TagBox["\[Psi]", PolyGamma] ( k + 1 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) + sin ( - 1 4 z ) Si ( 2 - 1 4 z ) ) ) z 2 k + ( - 1 ) 5 / 8 2 1 2 - n - 1 4 ( n π ) z 1 2 - n π n ! k = 0 n - 1 2 2 2 k ( n 2 k + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( - 2 k + 2 n - 1 ) ! k ( 1 4 ( - 3 ) ( 2 n + 1 ) π ( - Chi ( 2 - 1 4 z ) sinh ( - 1 4 z ) - ( ψ TagBox["\[Psi]", PolyGamma] ( k + 3 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) sinh ( - 1 4 z ) + cosh ( - 1 4 z ) Shi ( 2 - 1 4 z ) ) + ( - 1 ) k ( Ci ( 2 - 1 4 z ) sin ( - 1 4 z ) + ( ψ TagBox["\[Psi]", PolyGamma] ( k + 3 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( k - n + 1 2 ) ) sin ( - 1 4 z ) - cos ( - 1 4 z ) Si ( 2 - 1 4 z ) ) ) z 2 k /; n TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition 1 0 Subscript BesselJ ber -1 n -1 1 2 z -1 3 4 -1 KelvinBei -1 n -1 1 2 z z -1 -1 1 4 z KelvinBer -1 n -1 1 2 z -1 3 8 2 -1 n -1 1 2 7 n 4 -1 z -1 n -1 1 2 1 2 n -1 k 0 n 2 -1 2 2 k Binomial n 2 k 2 n -1 2 k k 1 4 -3 2 n 1 -1 1 4 z CoshIntegral 2 -1 1 4 z -1 1 4 z PolyGamma k 1 2 -1 PolyGamma k -1 n 1 2 -1 -1 1 4 z SinhIntegral 2 -1 1 4 z -1 k -1 1 4 z CosIntegral 2 -1 1 4 z -1 1 4 z PolyGamma k 1 2 -1 PolyGamma k -1 n 1 2 -1 1 4 z SinIntegral 2 -1 1 4 z z 2 k -1 5 8 2 1 2 -1 n -1 1 4 n z 1 2 -1 n 1 2 n -1 k 0 n -1 2 -1 2 2 k Binomial n 2 k 1 -2 k 2 n -1 k 1 4 -3 2 n 1 -1 CoshIntegral 2 -1 1 4 z -1 1 4 z -1 PolyGamma k 3 2 -1 PolyGamma k -1 n 1 2 -1 1 4 z -1 1 4 z SinhIntegral 2 -1 1 4 z -1 k CosIntegral 2 -1 1 4 z -1 1 4 z PolyGamma k 3 2 -1 PolyGamma k -1 n 1 2 -1 1 4 z -1 -1 1 4 z SinIntegral 2 -1 1 4 z z 2 k n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["KelvinBer", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List[RowBox[List["-", "n_"]], "-", FractionBox["1", "2"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["-", "3"]], ")"]], " ", "\[Pi]", " ", RowBox[List["KelvinBei", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["KelvinBer", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]], ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "8"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["7", " ", "\[ImaginaryI]", " ", "n", " ", "\[Pi]"]], "4"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]], " ", SuperscriptBox["\[ImaginaryI]", "k"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["-", "3"]], ")"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["1", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Cosh", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["Sinh", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["1", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Cos", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]], " ", RowBox[List["CosIntegral", "[", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["n", "!"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["5", "/", "8"]]], " ", SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "n", " ", "\[Pi]"]], ")"]]]]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "2"], "-", "n"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], "!"]], " ", SuperscriptBox["\[ImaginaryI]", "k"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["-", "3"]], ")"]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["3", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]]]], " ", RowBox[List["Sinh", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Cosh", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]], " ", RowBox[List["SinhIntegral", "[", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["Sinh", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]], " ", RowBox[List["CoshIntegral", "[", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", FractionBox["3", "2"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "n", "+", FractionBox["1", "2"]]], "]"]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]], " ", RowBox[List["CosIntegral", "[", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]]]], "-", RowBox[List[RowBox[List["Cos", "[", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["n", "!"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02