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'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 ''f''(''x'') = ''x''^''n'' 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== The concept of evenness or oddness is only defined for functions whose domain and range both have an [[additive inverse]]. This includes [[abelian group|additive groups]], all [[ring (algebra)|ring]]s, all [[field (mathematics)|field]]s, and all [[vector space]]s. Thus, for example, a real-valued function of a real variable could be even or odd, as could a complex-valued function of a vector variable, and so on. The examples are real-valued functions of a real variable, to illustrate the [[symmetry]] of their graphs. ===Even functions=== [[Image:Function x^2.svg|right|thumb|{{nowrap|''&fnof;''(''x'') {{=}} ''x''²}} is an example of an even function.]] Let ''f''(''x'') be a [[real number|real]]-valued function of a real variable. Then ''f'' is '''even''' if the following equation holds for all ''x'' and ''-x'' in the domain of ''f'': :<math> f(x) = f(-x). \, </math> Geometrically speaking, the graph face of an even function is [[symmetry|symmetric]] with respect to the ''y''-axis, meaning that its [[graph of a function|graph]] remains unchanged after [[reflection (mathematics)|reflection]] about the ''y''-axis. Examples of even functions are [[absolute value|{{!}}''x''{{!}}]], ''x''², ''x''⁴, [[trigonometric function|cos]](''x''), and [[hyperbolic function|cosh]](''x''). ===Odd functions=== [[Image:Function-x3.svg|right|thumb|{{nowrap|''&fnof;''(''x'') {{=}} ''x''³}} is an example of an odd function.]] Again, let ''f''(''x'') be a [[real number|real]]-valued function of a real variable. Then ''f'' is '''odd''' if the following equation holds for all ''x'' and ''-x'' in the domain of ''f'': :<math> -f(x) = f(-x), \, </math> or :<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 of a function|graph]] remains unchanged after [[coordinate rotation|rotation]] of 180 [[degree (angle)|degree]]s about the origin. Examples of odd functions are ''x'', ''x''³, [[sine|sin]](''x''), [[hyperbolic function|sinh]](''x''), and [[error function|erf]](''x''). ==Some facts== [[Image:Function-x3plus1.svg|right|thumb|{{nowrap|''&fnof;''(''x'') {{=}} ''x''³ + 1}} is neither even nor odd.]] 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. Properties involving Fourier series, Taylor series, derivatives and so on may only be used when they can be assumed to exist. ===Basic properties=== * The only function whose domain is all real numbers which is ''both'' even and odd is the [[constant function]] which is identically zero (i.e., ''f''(''x'') = 0 for all ''x'').<ref>For a description of the family of functions which are both odd and even, see http://studentpersonalpages.loyola.edu/zmpisano/www/</ref> * The sum of two even functions is even, and any constant multiple of an even function is even. * The sum of two odd functions is odd, and any constant multiple of an odd function is odd. * The difference between two odd functions is odd. * The difference between two even functions is even. * 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. * The [[derivative]] of an even function is odd. * The derivative of an odd function is even. * 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 either an odd or an even function with an even function is even (but not vice versa). * 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''). * 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). ===The sum of even and odd functions=== * Every function can be expressed as the sum of an even and an odd function. ''Proof'': Let <math>f{(x)}</math> be any function that is defined for all [[real number]]s. We can rewrite this as: <math> \frac{f{(x)}}{2}+\frac{f{(x)}}{2}+\frac{f{(-x)}}{2}-\frac{f{(-x)}}{2}</math>. In turn this can be rewritten as <math>\frac {f{(x)}+f{(-x)}}{2} + \frac {f{(x)}-f{(-x)}}{2} </math>. Let <math>g{(x)}</math> be <math>\frac {f{(x)}+f{(-x)}}{2}</math> and <math>h{(x)}</math> be <math>\frac {f{(x)}-f{(-x)}}{2}</math>. Clearly <math> f{(x)}=g{(x)}+h{(x)}</math>. Now <math>g{(x)}</math> is even <math>\because g{(-x)}=\frac {f{(-x)}+f{(x)}}{2}=g{(x)}</math>. <math>h{(x)}</math> is odd <math>\because h{(-x)}=\frac {f{(-x)}-f{(x)}}{2}=-\frac {f{(x)}-f{(-x)}}{2}=-h{(x)}</math>. [[Q.E.D.]] * The [[addition|sum]] of an even and odd function is neither even nor odd, unless one of the functions is equal to zero over the given [[Domain of a function|domain]]. ===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. ===Algebraic structure=== * 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-valued functions is the [[direct sum of vector spaces|direct sum]] of the [[linear subspace|subspace]]s of even and odd functions. In other words, every function ''f''(''x'') can be written uniquely as the sum of an even function and an odd function: :: :: <math>f(x)=f_\text{e}(x) + f_\text{o}(x)\, ,</math> : where :: <math>f_\text{e}(x) = \tfrac12[f(x)+f(-x)]</math> : is even and :: <math>f_\text{o}(x) = \tfrac12[f(x)-f(-x)]</math> : is odd. For example, if ''f'' is exp, then ''f''_e is cosh and ''f''_o is&nbsp;sinh. *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. ==Harmonics== In [[signal processing]], [[harmonic distortion]] occurs when a [[sine wave]] signal is sent through a memoryless [[nonlinear system]], that is, a system whose output at time <math>t</math> only depends on the input at time <math>t</math> 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 <math>f</math>:<ref>[http://www.uaudio.com/webzine/2005/october/content/content2.html Ask the Doctors: Tube vs. Solid-State Harmonics]</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. ==See also== *[[Hermitian function]] for a generalization in complex numbers *[[Taylor series]] *[[Fourier series]] *[[Holstein–Herring method]] ==Notes== <references/> [[Category:Calculus]] [[Category:Parity]] [[Category:Types of functions]]'