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{{Short description|
{{distinguish|Even and odd numbers}}
[[File:Sintay SVG.svg|thumb|The [[sine function]] and all of its [[Taylor polynomial]]s are odd functions
[[File:Développement limité du cosinus.svg|thumb|The [[cosine function]] and all of its [[Taylor polynomials]] are even functions
In [[mathematics]], an '''even function''' is a [[real 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.▼
▲
Even functions are those real functions whose [[graph of a function|graph]] is [[symmetry (geometry)|self-symmetric]] with respect to the {{nowrap|{{mvar|y}}-axis,}} and odd functions are those whose graph is self-symmetric with respect to the [[origin (mathematics)|origin]].
If the ___domain of a real function is self-symmetric with respect to the origin, then the function can be uniquely decomposed as the sum of an even function and an odd function.
==Early history==
The concept of even and odd functions appears to date back to the early 18th century, with [[Leonard Euler]] playing a significant role in their formalization. Euler introduced the concepts of even and odd functions (using Latin terms ''pares'' and ''impares'') in his work ''Traiectoriarum Reciprocarum Solutio'' from 1727. Before Euler, however, [[Isaac Newton]] had already developed geometric means of deriving coefficients of power series when writing the ''Principia'' (1687), and included algebraic techniques in an early draft of his ''Quadrature of Curves,'' though he removed it before publication in 1706. It is also noteworthy that Newton didn't explicitly name or focus on the even-odd decomposition, his work with power series would have involved understanding properties related to even and odd powers.
==Definition and examples==
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===Even functions===
[[Image:Function x^2.svg|right|thumb|<math>f(x)=x^2</math> is an example of an even function.]]
or equivalently
▲:<math>f(x) - f(-x) = 0.</math>
Geometrically, the graph of an even function is [[Symmetry|symmetric]] with respect to the ''y''-axis, meaning that its graph remains unchanged after [[Reflection (mathematics)|reflection]] about the ''y''-axis.
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*The [[absolute value]] <math>x \mapsto |x|,</math>
*<math>x \mapsto x^2,</math>
*<math>x \mapsto x^
*[[trigonometric function|cosine]] <math>\cos,</math>
*[[hyperbolic function|hyperbolic cosine]] <math>\cosh,</math>
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===Odd functions===
[[Image:Function-x3.svg|right|thumb|<math>f(x)=x^3</math> is an example of an odd function.]]
<math display =block>f(-x) = -f(x)</math>
or equivalently
<math display =block>f(x) + f(-x) = 0.</math>
Geometrically, the graph of an odd function has rotational symmetry with respect to the [[Origin (mathematics)|origin]], meaning that its graph remains unchanged after [[Rotation (mathematics)|rotation]] of 180 [[Degree (angle)|degree]]s about the origin.▼
If <math>x=0</math> is in the ___domain of an odd function <math>f(x)</math>, then <math>f(0)=0</math>.
▲:<math>f(x) + f(-x) = 0.</math>
▲Geometrically, the graph of an odd function has rotational symmetry with respect to the [[Origin (mathematics)|origin]], meaning that its graph remains unchanged after [[Rotation (mathematics)|rotation]] of 180 [[Degree (angle)|degree]]s about the origin.
Examples of odd functions are:
*The [[sign function]] <math>x \mapsto \sgn(x),</math>
*The identity function <math>x \mapsto x,</math>
*<math>x \mapsto x^
*<math>x \mapsto \sqrt[n]{x}</math> for any odd positive integer <math>n,</math>
*[[sine]] <math>\sin,</math>
*[[hyperbolic function|hyperbolic sine]] <math>\sinh,</math>
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===Multiplication and division===
* The [[multiplication|product]] and [[Division (mathematics)|quotient]] of two even functions is an even function.
**
**
* The product
* The
**
===Composition===
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* The composition of an even function and an odd function is even.
* The composition of any function with an even function is even (but not vice versa).
===Inverse function===
* If an odd function is [[inverse function|invertible]], then its inverse is also odd.
==Even–odd decomposition==
▲|equation = {{NumBlk||<math>f_\text{e}(x) = \frac {f(x)+f(-x)}{2}</math>|{{EquationRef|Eq.3}}}}
and
▲then <math>f_\text{e}</math> is even, <math>f_\text{o}</math> is odd, and
▲: <math>f(x)=f_\text{e}(x) + f_\text{o}(x).</math>
This decomposition is unique since, if
:<math>f(x)=g(x)+h(x),</math>
where {{mvar|g}} is even and {{mvar|h}} is odd, then <math>g=f_\text{
: <math>\begin{align}
2f_\text{e}(x) &=f(x)+f(-x)= g(x) + g(-x) +h(x) +h(-x) = 2g(x),\\
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For example, the [[hyperbolic cosine]] and the [[hyperbolic sine]] may be regarded as the even and odd parts of the exponential function, as the first one is an even function, the second one is odd, and
:<math>e^x=\underbrace{\cosh (x)}_{f_\text{
[[Joseph Fourier|Fourier]]'s [[sine and cosine transforms]] also perform even–odd decomposition by representing a function's odd part with [[sine waves]] (an odd function) and the function's even part with cosine waves (an even function).
==Further algebraic properties==
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A function's being odd or even does not imply [[differentiable function|differentiability]], or even [[continuous function|continuity]]. For example, the [[Dirichlet function]] is even, but is nowhere continuous.
In the following, properties involving [[derivative]]s, [[Fourier series]], [[Taylor series]] are considered, and
===Basic analytic properties===
* The [[derivative]] of an even function is odd.
* The derivative of an odd function is even.
*
*:<math>\int_{-A}^{A} f(x)\,dx = 0</math>.
** This implies that the [[Cauchy principal value]] of an odd function over the entire real line is zero.
* If an even function is integrable over a bounded symmetric interval <math>[-A,A]</math>, the integral over that interval is twice the integral from 0 to ''A''; that is<ref>{{cite web|url=http://mathworld.wolfram.com/EvenFunction.html|title=Even Function|first=Weisstein, Eric|last=W.|website=mathworld.wolfram.com}}</ref>
*:<math>\int_{-A}^{A} f(x)\,dx = 2\int_{0}^{A} f(x)\,dx</math>.
** This property is also true for the [[improper integral]] when <math>A = \infty</math>, provided the integral from 0 to <math>\infty</math> converges.
===Series===
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** Simple examples are a half-wave rectifier, and clipping in an asymmetrical [[class-A amplifier]].
==Generalizations==
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===Complex-valued functions===
The definitions for even and odd symmetry for [[Complex number|complex-valued]] functions of a real argument are similar to the real case.
{{Cite book |last1=Oppenheim |first1=Alan V. |author-link=Alan V. Oppenheim |last2=Schafer |first2=Ronald W. |author2-link=Ronald W. Schafer |last3=Buck |first3=John R. |title=Discrete-time signal processing |year=1999 |publisher=Prentice Hall |___location=Upper Saddle River, N.J. |isbn=0-13-754920-2 |edition=2nd |page=55
}}</ref><ref name=ProakisManolakis/>
'''
A complex-valued function of a real argument <math>f: \mathbb{R} \to \mathbb{C}</math> is called ''
:<math>f(x)=\overline{f(-x)} \quad \text{for all } x \in \mathbb{R}</math>
A complex valued function is conjugate symmetric if and only if its [[real part]] is an even function and its [[imaginary part]] is an odd function.
A typical example of a conjugate symmetric function is the [[cis function]]
A complex-valued function of a real argument <math>f: \mathbb{R} \to \mathbb{C}</math> is called ''odd symmetric'' if:▼
:<math>x \to e^{ix}=\cos x + i\sin x</math>
'''Conjugate antisymmetry:'''
▲A complex-valued function of a real argument <math>f: \mathbb{R} \to \mathbb{C}</math> is called ''
:<math>f(x)=-\overline{f(-x)} \quad \text{for all } x \in \mathbb{R}</math>
A complex valued function is conjugate antisymmetric if and only if its [[real part]] is an odd function and its [[imaginary part]] is an even function.
===Finite length sequences===
The definitions of odd and even symmetry are extended to ''N''-point sequences (i.e. functions of the form <math>f: \left\{0,1,\ldots,N-1\right\} \to \mathbb{R}</math>) as follows:<ref name=ProakisManolakis>{{Citation |
'''Even symmetry:'''
A ''N''-point sequence is called ''
:<math>f(n) = f(N-n) \quad \text{for all } n \in \left\{ 1,\ldots,N-1 \right\}.</math>
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'''Odd symmetry:'''
A ''N''-point sequence is called ''
:<math>f(n) = -f(N-n) \quad \text{for all } n \in \left\{1,\ldots,N-1\right\}. </math>
Such a sequence is sometimes called an '''anti-palindromic sequence'''; see also [[Palindromic polynomial|Antipalindromic polynomial]].
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==References==
*{{Citation |
[[Category:Calculus]]
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