Even and odd functions: Difference between revisions

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

{{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 [[functionreal (mathematics)|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==

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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.]]

Let ''f'' be aA [[real-valued function]] of a real variable. Then {{math|''f''}} is '''even''' if the following equation holds, for allevery ''{{mvar|x''}} suchin thatits ''x'' and___domain, {{math|−''x''}} areis also in theits ___domain of ''f'':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) = 0.</math>

or equivalently

{{Equation box 1

:<math display=block>f(x) +- f(-x) = 0.</math>

|indent =

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|equation = {{NumBlk||<math>f(-x) = f(x)</math>|{{EquationRef|Eq.1}}}}

|cellpadding= 6

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|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.

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*The [[absolute value]] <math>x \mapsto |x|,</math>

*<math>x \mapsto x^2,</math>

*<math>x \mapsto x^4n</math> for any even integer <math>n,</math>

*[[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.]]

Again, let ''f'' be aA real-valued function of a real variable. Then {{math|''f''}} is '''odd''' if the following equation holds, for allevery ''{{mvar|x''}} suchin thatits ''x'' and___domain, {{math|−''x''}} areis also in theits ___domain of ''f'':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.

{{Equation box 1

|indent =

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|equation = {{NumBlk||<math>f(-x) = -f(x)</math>|{{EquationRef|Eq.2}}}}

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|background colour=#F5FFFA}}

If <math>x=0</math> is in the ___domain of an odd function <math>f(x)</math>, then <math>f(0)=0</math>.

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 [[sign function]] <math>x \mapsto \sgn(x),</math>

*The identity function <math>x \mapsto x,</math>

*<math>x \mapsto x^3n</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>

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===Multiplication and division===

* The [[multiplication|product]] and [[Division (mathematics)|quotient]] of two even functions is an even function.

** ThatThis implies that the product of any number of even functions is analso even function as well.

** TheThis productimplies ofthat twothe odd[[reciprocal functionsfunction|reciprocal]] isof an even function is also even.

* The product ofand anquotient evenof function and antwo odd functionfunctions is an oddeven function.

* The [[Divisionproduct (mathematics)|quotient]]and both quotients of twoan even functionsfunction and an odd function is an evenodd function.

** TheThis quotientimplies ofthat twothe oddreciprocal functions isof an evenodd function is odd.

* The quotient of an even function and an odd function is an odd function.

===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==

EveryIf 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;, ifand are onedefined definesby

|equation<math display= {{NumBlk||<math"block">f_\text{eeven}(x) = \frac {f(x)+f(-x)}{2},</math>|{{EquationRef|Eq.3}}}}

{{Equation box 1

|indent =

|title=

|equation = {{NumBlk||<math>f_\text{e}(x) = \frac {f(x)+f(-x)}{2}</math>|{{EquationRef|Eq.3}}}}

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|border colour = #0073CF

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and

: <math>f(x) display=block>f_\text{eodd}(x) += f_\textfrac {o}f(x)-f(-x)}{2}.</math>

thenIt is straightforward to verify that <math>f_\text{eeven}</math> is even, <math>f_\text{oodd}</math> is odd, and <math>f=f_\text{even}+f_\text{odd}.</math>

{{Equation box 1

|indent =

|title=

|equation = {{NumBlk||<math>f_\text{o}(x) = \frac {f(x)-f(-x)}{2}</math>|{{EquationRef|Eq.4}}}}

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|border colour = #0073CF

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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

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{eeven}</math> and <math>h=f_\text{oodd},</math> since

: <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{eeven}(x)} + \underbrace{\sinh (x)}_{f_\text{oodd}(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==

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* The [[derivative]] of an even function is odd.

* The derivative of an odd function is even.

* The [[integral]] ofIf an odd function fromis −''A''[[integral|integrable]] toover +''A''a is zero[[Interval (where ''A'' is finite, and the function has no vertical asymptotes between −''A'' and ''A''mathematics). For an odd function that is integrable over a|bounded 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>.

** This implies that the [[Cauchy principal value]] of an odd function over the entire real line is zero.

* 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); that is

* 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]].

Note that thisThis 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==

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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. butIn involve[[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/>

'''EvenConjugate symmetry:'''

A complex-valued function of a real argument <math>f: \mathbb{R} \to \mathbb{C}</math> is called ''evenconjugate 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.

'''Odd symmetry:'''

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 ''oddconjugate symmetricantisymmetric'' 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 | lastlast1 =Proakis | firstfirst1 =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 ''evenconjugate symmetric'' if

:<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 ''oddconjugate symmetricantisymmetric'' 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]].

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==References==

*{{Citation |lastlast1=Gelfand |firstfirst1=I. M. |last2=Glagoleva |first2=E. G. |last3=Shnol |first3=E. E. |author-linklink1=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]]