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Exp






Mathematica Notation

Traditional Notation









Elementary Functions > Exp[z] > Transformations > Transformations and argument simplifications > Argument involving inverse trigonometric and hyperbolic functions > Argument involving symbolic multiples of inverse trigonometric and hyperbolic functions





http://functions.wolfram.com/01.03.16.0065.01









  


  










Input Form





E^(((I n)/2) ArcSec[z]) == n Sum[(((-1)^k (n - k - 1)! 2^(n/2 - k - 1))/(k! (n - 2 k)!)) ((z + 1)/z)^(n/2 - k), {k, 0, Floor[n/2]}] + I ((z + 1)/z)^((n - 1)/2) Sqrt[(z - 1)/z] Sum[(-1)^k Binomial[-k + n - 1, k] 2^(n/2 - k - 1) (z/(z + 1))^k, {k, 0, Floor[(n - 1)/2]}] /; Element[n, Integers] && n > 0










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "n_"]], ")"]], " ", RowBox[List["ArcSec", "[", "z_", "]"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["n", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox["2", RowBox[List[FractionBox["n", "2"], "-", "k", "-", "1"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "z"], ")"]], RowBox[List[FractionBox["n", "2"], "-", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], "!"]]]]]]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["z", "+", "1"]], "z"], ")"]], FractionBox[RowBox[List["n", "-", "1"]], "2"]], " ", SqrtBox[FractionBox[RowBox[List["z", "-", "1"]], "z"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "+", "n", "-", "1"]], ",", "k"]], "]"]], " ", SuperscriptBox["2", RowBox[List[FractionBox["n", "2"], "-", "k", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", RowBox[List["z", "+", "1"]]], ")"]], "k"]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2001-10-29





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