Exponential function: Representations through more general functions (formula 01.03.26.0118)

Exp

$Mathematica Notation$

Elementary Functions

Exp[z]

Representations through more general functions

Through Meijer G

Classical cases for the direct function itself

http://functions.wolfram.com/01.03.26.0118.01

Input Form

E^z^(1/n) == 2^((n - 1)/2) Sqrt[n] Pi^((1 + n)/2) MeijerG[{{}, {1/(2 n), 3/(2 n), 5/(2 n), \[Ellipsis], (2 n - 1)/(2 n)}}, {{0, 1/n, 2/n, \[Ellipsis], (n - 1)/n}, {1/(2 n), 3/(2 n), 5/(2 n), \[Ellipsis], (2 n - 1)/(2 n)}}, z/n^n] /; 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]", SuperscriptBox["z_", FractionBox["1", "n_"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", FractionBox[RowBox[List["n", "-", "1"]], "2"]], " ", SqrtBox["n"], " ", SuperscriptBox["\[Pi]", FractionBox[RowBox[List["1", "+", "n"]], "2"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", RowBox[List["2", " ", "n"]]], ",", FractionBox["3", RowBox[List["2", " ", "n"]]], ",", FractionBox["5", RowBox[List["2", " ", "n"]]], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], RowBox[List["2", " ", "n"]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "n"], ",", FractionBox["2", "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["n", "-", "1"]], "n"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", RowBox[List["2", " ", "n"]]], ",", FractionBox["3", RowBox[List["2", " ", "n"]]], ",", FractionBox["5", RowBox[List["2", " ", "n"]]], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], RowBox[List["2", " ", "n"]]]]], "}"]]]], "}"]], ",", FractionBox["z", SuperscriptBox["n", "n"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

2007-05-02