Even and odd functions: Difference between revisions

Revision as of 09:36, 22 January 2024

Revision as of 10:36, 22 January 2024

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{{Short description|Mathematical functions with specific symmetries}}

{{distinguish|Even and odd numbers}}

[[File:Sintay SVG.svg|thumb|The [[sine function]] and all of its [[Taylor polynomial]]s are odd functions. This image shows <math>\sin(x)</math> and its Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.]]

[[File:Développement limité du cosinus.svg|thumb|The [[cosine function]] and all of its [[Taylor polynomials]] are even functions. This image shows <math>\cos(x)</math> and its Taylor approximation of degree 4.∮]]

In [[mathematics]], an '''even function''' is a [[function (mathematics)|function]] such that <math>f(-x)=f(x)</math> for every <math>x</math> in its [[___domain of a function|___domain]]. Similarly, an '''odd function''' is a function such that <math>f(-x)=-f(x)</math> for every <math>x</math> in its ___domain.

In [[mathematics]], '''even functions''' and '''odd functions''' are [[function (mathematics)|function]]s which satisfy particular [[Symmetry in mathematics|symmetry]] relations, with respect to taking [[additive inverse]]s. They are important in many areas of [[mathematical analysis]], especially the theory of [[power series]] and [[Fourier series]]. They are named for the [[parity (mathematics)|parity]] of the powers of the [[Power Function|power functions]] which satisfy each condition: the function <math>f(x) = x^n</math> is an even function if ''n'' is an even [[integer]], and it is an odd function if ''n'' is an odd integer.

In [[mathematics]], '''even functions''' and '''odd functions''' are [[function (mathematics)|function]]s which satisfy particular [[Symmetry in mathematics|symmetry]] relations, with respect to taking [[additive inverse]]s. They are important in many areas of [[mathematical analysis]], especially the theory of [[power series]] and [[Fourier series]]. They are named for the [[parity (mathematics)|parity]] of the powers of the [[Power Function|power functions]] which satisfy each condition: the function <math>f(x) = x^n</math> is an even function if ''n'' is an [[even [[integer]], and it is an odd function if ''n'' is an odd integer.

==Definition and examples==