Editing Even and odd functions

{{Short description|Functions such that f(–x) equals f(x) or –f(x)}}
{{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.

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 even if ''n'' is an [[even integer]], and it is odd 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==

Evenness and oddness are generally considered for [[real function]]s, that is real-valued functions of a real variable. However, the concepts may be more generally defined for functions whose [[domain of a function|domain]] and [[codomain]] both have a notion of [[additive inverse]]. This includes [[abelian group]]s, all [[Ring (algebra)|rings]], all [[Field (mathematics)|fields]], and all [[vector space]]s. Thus, for example, a real function could be odd or even (or neither), as could a [[Complex number|complex]]-valued function of a vector variable, and so on.

The given examples are real functions, to illustrate the [[symmetry]] of their [[Graph of a function|graphs]].

===Even functions===
[[Image:Function x^2.svg|right|thumb|<math>f(x)=x^2</math> is an example of an even function.]]
A [[real function]] {{math|''f''}} is '''even''' if, for every {{mvar|x}} in its domain, {{math|−''x''}} is also in its domain and<ref name=FunctionsAndGraphs>{{cite book|first1=I. M.|last1=Gel'Fand|author1-link=Israel Gelfand|first2=E. G.|last2=Glagoleva|author2-link=E. G. Glagoleva|first3=E. E.|last3=Shnol|title=Functions and Graphs|year=1990|publisher=Birkhäuser|isbn=0-8176-3532-7|url-access=registration|url=https://archive.org/details/functionsgraphs0000gelf}}</ref>{{rp|p. 11}}
<math display=block>f(-x) = f(x)</math>
or equivalently
<math display=block>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.

Examples of even functions are:
*The [[absolute value]] <math>x \mapsto |x|,</math>
*<math>x \mapsto x^2,</math>
*<math>x \mapsto x^n</math> for any even integer <math>n,</math>
*[[trigonometric function|cosine]] <math>\cos,</math>
*[[hyperbolic function|hyperbolic cosine]] <math>\cosh,</math>
*[[Gaussian function]] <math>x \mapsto \exp (-x^2). </math>

===Odd functions===
[[Image:Function-x3.svg|right|thumb|<math>f(x)=x^3</math> is an example of an odd function.]]
A real function {{math|''f''}} is '''odd''' if, for every {{mvar|x}} in its domain, {{math|−''x''}} is also in its domain and<ref name=FunctionsAndGraphs/>{{rp|p. 72}}
<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>.

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^n</math> for any odd integer <math>n,</math>
*<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>
*The [[error function]] <math>\operatorname{erf}.</math>

[[Image:Function-x3plus1.svg|right|thumb|<math>f(x)=x^3+1</math> is neither even nor odd.]]

==Basic properties==

===Uniqueness===
* If a function is both even and odd, it is equal to 0 everywhere it is defined.
* If a function is odd, the [[absolute value]] of that function is an even function.

===Addition and subtraction===
* The [[addition|sum]] of two even functions is even.
* The sum of two odd functions is odd.
* The [[subtraction|difference]] between two odd functions is odd.
* The difference between two even functions is even.
* The sum of an even and odd function is not even or odd, unless one of the functions is equal to zero over the given [[Domain of a function|domain]].

===Multiplication and division===
* The [[multiplication|product]] and [[Division (mathematics)|quotient]] of two even functions is an even function.
** This implies that the product of any number of even functions is also even.
** This implies that the [[reciprocal function|reciprocal]] of an even function is also even.
* The product and quotient of two odd functions is an even function.
* The product and both quotients of an even function and an odd function is an odd function.
** This implies that the reciprocal of an odd function is odd.

===Composition===
* The [[function composition|composition]] of two even functions is even.
* The composition of two odd functions is odd.
* 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==
If a real function has a domain that is self-symmetric with respect to the origin, it may be uniquely decomposed as the sum of an even and an odd function, which are called respectively the '''even part''' (or the '''even component''') and the '''odd part''' (or the '''odd component''') of the function, and are defined by
<math display="block">f_\text{even}(x) = \frac {f(x)+f(-x)}{2},</math>
and
<math display=block>f_\text{odd}(x) = \frac {f(x)-f(-x)}{2}.</math>

It is straightforward to verify that <math>f_\text{even}</math> is even, <math>f_\text{odd}</math> is odd, and <math>f=f_\text{even}+f_\text{odd}.</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{even}</math> and <math>h=f_\text{odd},</math>  since
: <math>\begin{align}
2f_\text{e}(x) &=f(x)+f(-x)= g(x) + g(-x) +h(x) +h(-x) = 2g(x),\\
2f_\text{o}(x) &=f(x)-f(-x)= g(x) - g(-x) +h(x) -h(-x) = 2h(x).
\end{align}</math>

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{even}(x)} + \underbrace{\sinh (x)}_{f_\text{odd}(x)}</math>.
[[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==
* Any [[linear combination]] of even functions is even, and the even functions form a [[vector space]] over the [[Real number|real]]s. Similarly, any linear combination of odd functions is odd, and the odd functions also form a vector space over the reals. In fact, the vector space of ''all'' real functions is the [[Direct sum of vector spaces|direct sum]] of the [[Linear subspace|subspaces]] of even and odd functions. This is a more abstract way of expressing the property in the preceding section.
**The space of functions can be considered a [[graded algebra]] over the real numbers by this property, as well as some of those above.

*The even functions form a [[algebra over a field|commutative algebra]] over the reals. However, the odd functions do ''not'' form an algebra over the reals, as they are not [[Closure (mathematics)|closed]] under multiplication.

==Analytic properties==

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 these concepts are thus supposed to be defined for the considered functions.

===Basic analytic properties===
* The [[derivative]] of an even function is odd. 
* The derivative of an odd function is even.
* If an odd function is [[integral|integrable]] over a [[Interval (mathematics)|bounded symmetric interval]] <math>[-A,A]</math>, the integral over that interval is zero; that is<ref>{{cite web|url=http://mathworld.wolfram.com/OddFunction.html|title=Odd Function|first=Weisstein, Eric|last=W.|website=mathworld.wolfram.com}}</ref>
*:<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===
* The [[Maclaurin series]] of an even function includes only even powers.
* The Maclaurin series of an odd function includes only odd powers.
* The [[Fourier series]] of a [[periodic function|periodic]] even function includes only [[trigonometric function|cosine]] terms.
* The Fourier series of a periodic odd function includes only [[trigonometric function|sine]] terms.
*The [[Fourier transform]] of a purely real-valued even function is real and even. (see {{slink|Fourier_analysis|Symmetry_properties}})
*The Fourier transform of a purely real-valued odd function is imaginary and odd. (see {{slink|Fourier_analysis|Symmetry_properties}})

==Harmonics==
In [[signal processing]], [[harmonic distortion]] occurs when a [[sine wave]] signal is sent through a memory-less [[nonlinear system]], that is, a system whose output at time ''t'' only depends on the input at time ''t'' and does not depend on the input at any previous times. Such a system is described by a response function <math>V_\text{out}(t) = f(V_\text{in}(t))</math>. The type of [[harmonic]]s produced depend on the response function ''f'':<ref>{{Cite web|url=http://www.uaudio.com/webzine/2005/october/content/content2.html|title=Ask the Doctors: Tube vs. Solid-State Harmonics|last=Berners|first=Dave|date=October 2005|website=UA WebZine|publisher=Universal Audio|access-date=2016-09-22|quote=To summarize, if the function f(x) is odd, a cosine input will produce no even harmonics. If the function f(x) is even, a cosine input will produce no odd harmonics (but may contain a DC component). If the function is neither odd nor even, all harmonics may be present in the output.}}</ref>
* When the response function is even, the resulting signal will consist of only even harmonics of the input sine wave; <math>0f, 2f, 4f, 6f, \dots </math>
** The [[fundamental frequency|fundamental]] is also an odd harmonic, so will not be present.
** A simple example is a [[full-wave rectifier]].
** The <math>0f</math> component represents the DC offset, due to the one-sided nature of even-symmetric transfer functions.
* When it is odd, the resulting signal will consist of only odd harmonics of the input sine wave; <math>1f, 3f, 5f, \dots </math>
** The output signal will be half-wave [[symmetric]].
** A simple example is [[clipping (audio)|clipping]] in a symmetric [[Electronic amplifier|push-pull amplifier]].
* When it is asymmetric, the resulting signal may contain either even or odd harmonics; <math>1f, 2f, 3f, \dots </math>
** Simple examples are a half-wave rectifier, and clipping in an asymmetrical [[class-A amplifier]].

This does not hold true for more complex waveforms.  A [[sawtooth wave]] contains both even and odd harmonics, for instance.  After even-symmetric full-wave rectification, it becomes a [[triangle wave]], which, other than the DC offset, contains only odd harmonics.

==Generalizations==
===Multivariate functions===

'''Even symmetry:'''

A function <math>f: \mathbb{R}^n \to \mathbb{R} </math> is called ''even symmetric'' if:
:<math>f(x_1,x_2,\ldots,x_n)=f(-x_1,-x_2,\ldots,-x_n) \quad \text{for all } x_1,\ldots,x_n \in \mathbb{R}</math>

'''Odd symmetry:'''

A function <math>f: \mathbb{R}^n \to \mathbb{R} </math> is called ''odd symmetric'' if:
:<math>f(x_1,x_2,\ldots,x_n)=-f(-x_1,-x_2,\ldots,-x_n) \quad \text{for all } x_1,\ldots,x_n \in \mathbb{R}</math>

===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. In [[signal processing]], a similar symmetry is sometimes considered, which involves [[complex conjugation]].<ref name=Oppenheim>
{{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/>

'''Conjugate symmetry:'''

A complex-valued function of a real argument <math>f: \mathbb{R} \to \mathbb{C}</math> is called ''conjugate symmetric'' if
:<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]]
:<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 ''conjugate antisymmetric'' if:
:<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 | last1 =Proakis | first1 =John G. | last2 =Manolakis | first2 =Dimitri G. | title =Digital Signal Processing: Principles, Algorithms and Applications | place =Upper Saddle River, NJ | publisher =Prentice-Hall International | year =1996 | edition =3 | language =en | id =sAcfAQAAIAAJ | isbn =9780133942897 | url-access =registration | url =https://archive.org/details/digitalsignalpro00proa }}</ref>{{rp|p. 411}}

'''Even symmetry:'''

A ''N''-point sequence is called ''conjugate symmetric'' if
:<math>f(n) = f(N-n) \quad \text{for all } n \in \left\{ 1,\ldots,N-1 \right\}.</math>

Such a sequence is often called a '''palindromic sequence'''; see also [[Palindromic polynomial]].

'''Odd symmetry:'''

A ''N''-point sequence is called ''conjugate antisymmetric'' if
:<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]].

==See also==
*[[Hermitian function]] for a generalization in complex numbers
*[[Taylor series]]
*[[Fourier series]]
*[[Holstein–Herring method]]
*[[Parity (physics)]]

==Notes==
{{reflist}}

==References==
*{{Citation |last1=Gelfand |first1=I. M. |last2=Glagoleva |first2=E. G. |last3=Shnol |first3=E. E. |author-link1=Israel Gelfand |year=2002 | orig-year=1969 |title=Functions and Graphs |publisher=Dover Publications |publication-place=Mineola, N.Y |url=http://store.doverpublications.com/0486425649.html }}

[[Category:Calculus]]
[[Category:Parity (mathematics)]]
[[Category:Types of functions]]