Base of the natural logarithm: Representations through equivalent functions (formula 02.05.27.0003)

$Mathematica Notation$

Constants

Representations through equivalent functions

http://functions.wolfram.com/02.05.27.0003.01

Input Form

E^(Pi I k) == (-1)^k /; Element[k, Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "k"]]], "\[Equal]", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"]]], "/;", RowBox[List["k", "\[Element]", "Integers"]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> <mo> ⩵ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <power /> <exponentiale /> <apply> <times /> <pi /> <imaginaryi /> <ci> k </ci> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <in /> <ci> k </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox["\[ExponentialE]", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "k_"]]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], "/;", RowBox[List["k", "\[Element]", "Integers"]]]]]]]]

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

2001-10-29