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'{{Short description|1 = Mathematical functions such that f(-x) = f(x) (even) or f(-x) = -f(x) (odd)}} [[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]], '''even functions''' and '''odd functions''' are [[function (mathematics)|function]]s which satisfy particular [[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]]s 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== 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, 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.]] Let ''f'' be a real-valued function of a real variable. Then ''f'' is '''even''' if the following equation holds for all ''x'' such that ''x'' and ''-x'' in the domain of ''f'':<ref name=FunctionsAndGraphs>{{cite book|first1=I.M.|last1=Gel'Fand|first2=E.G.|last2=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}} {{Equation box 1 |indent = |title= |equation = {{NumBlk||<math>f(x) = f(-x)</math>|{{EquationRef|Eq.1}}}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}} or equivalently if the following equation holds for all such ''x'': :<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. Examples of even functions are: *The [[absolute value]] <math>x \mapsto |x|,</math> *<math>x \mapsto x^2,</math> *<math>x \mapsto x^4,</math> *[[trigonometric function|cosine]] <math>\cos,</math> *[[hyperbolic function|hyperbolic cosine]] <math>\cosh.</math> ===Odd functions=== [[Image:Function-x3.svg|right|thumb|<math>f(x)=x^3</math> is an example of an odd function.]] Again, let ''f'' be a real-valued function of a real variable. Then ''f'' is '''odd''' if the following equation holds for all ''x'' such that ''x'' and ''-x'' are in the domain of ''f'':<ref name=FunctionsAndGraphs/>{{rp|p. 72}} {{Equation box 1 |indent = |title= |equation = {{NumBlk||<math>-f(x) = f(-x)</math>|{{EquationRef|Eq.2}}}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}} or equivalently if the following equation holds for all such ''x'': :<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 identity function <math>x \mapsto x,</math> *<math>x \mapsto x^3,</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]] of two even functions is an even function. * The product of two odd functions is an even function. * The product of an even function and an odd function is an odd function. * The [[Division (mathematics)|quotient]] of two even functions is an even function. * The quotient of two odd functions is an even function. * The quotient of an even function and an odd function is an odd function. ===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). ==Even–odd decomposition== Every function may be uniquely decomposed as the sum of an even and an odd function, which are called respectively the '''even part''' and the '''odd part''' of the function; if one defines {{Equation box 1 |indent = |title= |equation = {{NumBlk||<math>f_\text{e}(x) = \frac {f(x)+f(-x)}{2}</math>|{{EquationRef|Eq.3}}}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}} and {{Equation box 1 |indent = |title= |equation = {{NumBlk||<math>f_\text{o}(x) = \frac {f(x)-f(-x)}{2}</math>|{{EquationRef|Eq.4}}}} |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}} 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> Conversely, if :<math>f(x)=g(x)+h(x),</math> where {{mvar|g}} is even and {{mvar|h}} is odd, then <math>g=f_\text{e}</math> and <math>h=f_\text{o},</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{e}(x)} + \underbrace{\sinh (x)}_{f_\text{o}(x)}</math>. ==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]], and so on suppose that these concepts are defined of the functions that are considered. ===Basic analytic properties=== * The [[derivative]] of an even function is odd. * The derivative of an odd function is even. * The [[integral]] of an odd function from &minus;''A'' to +''A'' is zero (where ''A'' is finite, and the function has no vertical asymptotes between &minus;''A'' and ''A''). For an odd function that is integrable over a symmetric interval, e.g. <math>[-A,A]</math>, the result of 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>. * The integral of an even function from &minus;''A'' to +''A'' is twice the integral from 0 to +''A'' (where ''A'' is finite, and the function has no vertical asymptotes between &minus;''A'' and ''A''. This also holds true when ''A'' is infinite, but only if the integral converges); that is ::<math>\int_{-A}^{A} f(x)\,dx = 2\int_{0}^{A} f(x)\,dx</math>. * The integral from &minus;''A'' to +''A'' of an even function divided by an even function to the power of an odd function plus one is the integral from 0 to +''A'' of the even function. <ref>{{cite web|url=https://www.youtube.com/watch?v=xiIsPEqyTqU|title=The Single Most Overpowered Integration Technique in Existence.|first=Maths|last=Flammable|website=youtube.com}}</ref> This means that : ::<math>\int_{-A}^{A} \frac{e(x)}{1+t(x)^{o(x)}}\, dx = \int_0^A e(x)\, dx</math> (where {{math|''e(x)'' and ''t(x)'' are even and ''o(x)'' is odd}}) ===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]]. Note that 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 but involve [[complex conjugation]]. '''Even symmetry:''' A complex-valued function of a real argument <math>f: \mathbb{R} \to \mathbb{C}</math> is called ''even symmetric'' if: :<math>f(x)=\overline{f(-x)} \quad \text{for all } x \in \mathbb{R}</math> '''Odd symmetry:''' A complex-valued function of a real argument <math>f: \mathbb{R} \to \mathbb{C}</math> is called ''odd symmetric'' if: :<math>f(x)=-\overline{f(-x)} \quad \text{for all } x \in \mathbb{R}</math> ===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 | last =Proakis | first =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 ''even 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 ''odd 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 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 |last=Gelfand |first=I. M. |last2=Glagoleva |first2=E. G. |last3=Shnol |first3=E. E. |author-link=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]]'