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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''<sup>''n''</sup> is an even function if ''n'' is an even integer, and it is an odd function if ''n'' is an odd integer.
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| ==Definition and examples==
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| 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.
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| The examples are real-valued functions of a real variable, to illustrate the [[symmetry]] of their graphs.
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| ===Even functions===
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| [[Image:Function x^2.svg|right|thumb|{{nowrap|''ƒ''(''x'') {{=}} ''x''<sup>2</sup>}} is an example of an even function.]]
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| 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'':
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| :<math>
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| f(x) = f(-x). \,
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| </math>
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| 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.
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| Examples of even functions are [[absolute value|{{!}}''x''{{!}}]], ''x''<sup>2</sup>, ''x''<sup>4</sup>, [[trigonometric function|cos]](''x''), and [[hyperbolic function|cosh]](''x'').
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| ===Odd functions===
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| [[Image:Function-x3.svg|right|thumb|{{nowrap|''ƒ''(''x'') {{=}} ''x''<sup>3</sup>}} is an example of an odd function.]]
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| 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'':
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| :<math>
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| -f(x) = f(-x), \,
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| </math>
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| or
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| :<math>
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| f(x) + f(-x) = 0. \,
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| </math>
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| 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.
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| Examples of odd functions are ''x'', ''x''<sup>3</sup>, [[sine|sin]](''x''), [[hyperbolic function|sinh]](''x''), and [[error function|erf]](''x'').
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| ==Some facts==
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| [[Image:Function-x3plus1.svg|right|thumb|{{nowrap|''ƒ''(''x'') {{=}} ''x''<sup>3</sup> + 1}} is neither even nor odd.]]
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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. Properties involving Fourier series, Taylor series, derivatives and so on may only be used when they can be assumed to exist.
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| ===Basic properties===
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| * 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>
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| * The sum of two even functions is even, and any constant multiple of an even function is even.
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| * The sum of two odd functions is odd, and any constant multiple of an odd function is odd.
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| * The difference between two odd functions is odd.
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| * The difference between two even functions is even.
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| * The [[multiplication|product]] of two even functions is an even function.
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| * The product of two odd functions is an even function.
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| * The product of an even function and an odd function is an odd function.
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| * The [[Division (mathematics)|quotient]] of two even functions is an even function.
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| * The quotient of two odd functions is an even function.
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| * The quotient of an even function and an odd function is an odd function.
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| * The [[derivative]] of an even function is odd.
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| * The derivative of an odd function is even.
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| * The [[function composition|composition]] of two even functions is even.
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| * The composition of two odd functions is odd.
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| * The composition of an even function and an odd function is even.
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| * The composition of either an odd or an even function with an even function is even (but not vice versa).
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| * The [[integral]] of an odd function from −''A'' to +''A'' is zero (where ''A'' is finite, and the function has no vertical asymptotes between −''A'' and ''A'').
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| * The integral of an even function from −''A'' to +''A'' is twice the integral from 0 to +''A'' (where ''A'' is finite, and the function has no vertical asymptotes between −''A'' and ''A''. This also holds true when ''A'' is infinite, but only if the integral converges).
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| ===The sum of even and odd functions===
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| * Every function can be expressed as the sum of an even and an odd function. ''Proof'':
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| Let <math>f{(x)}</math> be any function that is defined for all [[real number]]s.
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| We can rewrite this as:
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| <math> \frac{f{(x)}}{2}+\frac{f{(x)}}{2}+\frac{f{(-x)}}{2}-\frac{f{(-x)}}{2}</math>.
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| In turn this can be rewritten as <math>\frac {f{(x)}+f{(-x)}}{2} + \frac {f{(x)}-f{(-x)}}{2} </math>.
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| 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>.
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| Clearly <math> f{(x)}=g{(x)}+h{(x)}</math>.
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| Now <math>g{(x)}</math> is even <math>\because g{(-x)}=\frac {f{(-x)}+f{(x)}}{2}=g{(x)}</math>.
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| <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.]]
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| * 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]].
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| ===Series===
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| * The [[Maclaurin series]] of an even function includes only even powers.
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| * The Maclaurin series of an odd function includes only odd powers.
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| * The [[Fourier series]] of a [[periodic function|periodic]] even function includes only [[trigonometric function|cosine]] terms.
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| * The Fourier series of a periodic odd function includes only [[trigonometric function|sine]] terms.
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| ===Algebraic structure===
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| * 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:
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| ::
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| :: <math>f(x)=f_\text{e}(x) + f_\text{o}(x)\, ,</math>
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| : where
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| :: <math>f_\text{e}(x) = \tfrac12[f(x)+f(-x)]</math>
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| : is even and
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| :: <math>f_\text{o}(x) = \tfrac12[f(x)-f(-x)]</math>
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| : is odd. For example, if ''f'' is exp, then ''f''<sub>e</sub> is cosh and ''f''<sub>o</sub> is sinh.
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| *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.
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| ==Harmonics==
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| 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>
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| * 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>
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| ** The [[fundamental frequency|fundamental]] is also an odd harmonic, so will not be present.
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| ** A simple example is a [[full-wave rectifier]].
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| ** The <math>0f</math> component represents the DC offset, due to the one-sided nature of even-symmetric transfer functions.
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| * When it is odd, the resulting signal will consist of only odd harmonics of the input sine wave; <math>1f, 3f, 5f, \dots \ </math>
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| ** The output signal will be half-wave [[symmetric]].
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| ** A simple example is [[clipping (audio)|clipping]] in a symmetric [[Electronic amplifier|push-pull amplifier]].
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| * When it is asymmetric, the resulting signal may contain either even or odd harmonics; <math>1f, 2f, 3f, \dots \ </math>
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| ** Simple examples are a half-wave rectifier, and clipping in an asymmetrical [[class A amplifier]].
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| 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.
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| ==See also==
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| *[[Hermitian function]] for a generalization in complex numbers
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| *[[Taylor series]]
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| *[[Fourier series]]
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| *[[Holstein–Herring method]]
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| ==Notes==
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| <references/>
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| [[Category:Calculus]]
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| [[Category:Parity]]
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| [[Category:Types of functions]]
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