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Even and odd functions

From Wikipedia, the free encyclopedia

Functions such that f(–x) equals f(x) or –f(x)

Not to be confused withEven and odd numbers.

Thesine function and all of itsTaylor polynomials are odd functions.

Thecosine function and all of itsTaylor polynomials are even functions.

Inmathematics, aneven function is areal function such that $f(-x)=f(x)$ for every $x {\displaystyle x}$ in itsdomain. Similarly, anodd function is a function such that $f(-x)=-f(x)$ for every $x {\displaystyle x}$ in its domain.

They are named for theparity of the powers of thepower functions which satisfy each condition: the function $f(x)=x^{n}$ is even ifn is aneven integer, and it is odd ifn is an odd integer.

Even functions are those real functions whosegraph isself-symmetric with respect to they-axis, and odd functions are those whose graph is self-symmetric with respect to theorigin.

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, withLeonard Euler playing a significant role in their formalization. Euler introduced the concepts of even and odd functions (using Latin termspares andimpares) in his workTraiectoriarum Reciprocarum Solutio from 1727. Before Euler, however,Isaac Newton had already developed geometric means of deriving coefficients of power series when writing thePrincipia (1687), and included algebraic techniques in an early draft of hisQuadrature 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 forreal functions, that is real-valued functions of a real variable. However, the concepts may be more generally defined for functions whosedomain andcodomain both have a notion ofadditive inverse. This includesabelian groups, allrings, allfields, and allvector spaces. Thus, for example, a real function could be odd or even (or neither), as could acomplex-valued function of a vector variable, and so on.

The given examples are real functions, to illustrate thesymmetry of theirgraphs.

Even functions

$f(x)=x^{2}$ is an example of an even function.

{\displaystyle f(x)=x^{2}} — $f(x)=x^{2}$ is an example of an even function.

Areal functionf iseven if, for everyx in its domain,−x is also in its domain and^[1]^{: p. 11} $f(-x)=f(x)$ or equivalently $f(x)-f(-x)=0.$

Geometrically, the graph of an even function issymmetric with respect to they-axis, meaning that its graph remains unchanged afterreflection about they-axis.

Examples of even functions are:

Theabsolute value $x\mapsto |x|,$
$x\mapsto x^{2},$
$x\mapsto x^{n}$ for any even integer $n, {\displaystyle n,}$
cosine $\cos ,$
hyperbolic cosine $\cosh ,$
Gaussian function $x\mapsto \exp(-x^{2}).$

Odd functions

$f(x)=x^{3}$ is an example of an odd function.

{\displaystyle f(x)=x^{3}} — $f(x)=x^{3}$ is an example of an odd function.

A real functionf isodd if, for everyx in its domain,−x is also in its domain and^[1]^{: p. 72} $f(-x)=-f(x)$ or equivalently $f(x)+f(-x)=0.$

Geometrically, the graph of an odd function has rotational symmetry with respect to theorigin, meaning that its graph remains unchanged afterrotation of 180degrees about the origin.

If $x=0$ is in the domain of an odd function $f(x)$ , then $f(0)=0$ .

Examples of odd functions are:

Thesign function $x\mapsto \operatorname {sgn}(x),$
The identity function $x\mapsto x,$
$x\mapsto x^{n}$ for any odd integer $n, {\displaystyle n,}$
$x\mapsto {\sqrt[{n}]{x}}$ for any odd positive integer $n, {\displaystyle n,}$
sine $\sin ,$
hyperbolic sine $\sinh ,$
Theerror function $\operatorname {erf} .$

$f(x)=x^{3}+1$ is neither even nor odd.

{\displaystyle f(x)=x^{3}+1} — $f(x)=x^{3}+1$ 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, theabsolute value of that function is an even function.

Addition and subtraction

Thesum of two even functions is even.
The sum of two odd functions is odd.
Thedifference 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 givendomain.

Multiplication and division

Theproduct andquotient 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 thereciprocal 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

Thecomposition 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 isinvertible, 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 theeven part (or theeven component) and theodd part (or theodd component) of the function, and are defined by $f_{\text{even}}(x)={\frac {f(x)+f(-x)}{2}},$ and $f_{\text{odd}}(x)={\frac {f(x)-f(-x)}{2}}.$

It is straightforward to verify that $f_{\text{even}}$ is even, $f_{\text{odd}}$ is odd, and $f=f_{\text{even}}+f_{\text{odd}}.$

This decomposition is unique since, if

f(x)=g(x)+h(x),

whereg is even andh is odd, then $g=f_{\text{even}}$ and $h=f_{\text{odd}},$ since

{\begin{aligned}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{aligned}}

For example, thehyperbolic cosine and thehyperbolic 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

e^{x}=\underbrace {\cosh(x)} _{f_{\text{even}}(x)}+\underbrace {\sinh(x)} _{f_{\text{odd}}(x)}

.

Fourier'ssine and cosine transforms also perform even–odd decomposition by representing a function's odd part withsine waves (an odd function) and the function's even part with cosine waves (an even function).

Further algebraic properties

Anylinear combination of even functions is even, and the even functions form avector space over thereals. 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 ofall real functions is thedirect sum of thesubspaces 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 agraded algebra over the real numbers by this property, as well as some of those above.

The even functions form acommutative algebra over the reals. However, the odd functions donot form an algebra over the reals, as they are notclosed under multiplication.

Analytic properties

A function's being odd or even does not implydifferentiability, or evencontinuity. For example, theDirichlet function is even, but is nowhere continuous.

In the following, properties involvingderivatives,Fourier series,Taylor series are considered, and these concepts are thus supposed to be defined for the considered functions.

Basic analytic properties

Thederivative of an even function is odd.
The derivative of an odd function is even.
If an odd function isintegrable over abounded symmetric interval $[-A,A]$ , the integral over that interval is zero; that is^[2]
$\int _{-A}^{A}f(x)\,dx=0$ .
- This implies that theCauchy principal value of an odd function over the entire real line is zero.
If an even function is integrable over a bounded symmetric interval $[-A,A]$ , the integral over that interval is twice the integral from 0 toA; that is^[3]
$\int _{-A}^{A}f(x)\,dx=2\int _{0}^{A}f(x)\,dx$ .
- This property is also true for theimproper integral when $A=\infty$ , provided the integral from 0 to $\infty$ converges.

Series

TheMaclaurin series of an even function includes only even powers.
The Maclaurin series of an odd function includes only odd powers.
TheFourier series of aperiodic even function includes onlycosine terms.
The Fourier series of a periodic odd function includes onlysine terms.
TheFourier transform of a purely real-valued even function is real and even. (seeFourier analysis § Symmetry properties)
The Fourier transform of a purely real-valued odd function is imaginary and odd. (seeFourier analysis § Symmetry properties)

Harmonics

Insignal processing,harmonic distortion occurs when asine wave signal is sent through a memory-lessnonlinear system, that is, a system whose output at timet only depends on the input at timet and does not depend on the input at any previous times. Such a system is described by a response function $V_{\text{out}}(t)=f(V_{\text{in}}(t))$ . The type ofharmonics produced depend on the response functionf:^[4]

When the response function is even, the resulting signal will consist of only even harmonics of the input sine wave; $0f,2f,4f,6f,\dots$
- Thefundamental is also an odd harmonic, so will not be present.
- A simple example is afull-wave rectifier.
- The $0f$ 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; $1f,3f,5f,\dots$
- The output signal will be half-wavesymmetric.
- A simple example isclipping in a symmetricpush-pull amplifier.
When it is asymmetric, the resulting signal may contain either even or odd harmonics; $1f,2f,3f,\dots$
- Simple examples are a half-wave rectifier, and clipping in an asymmetricalclass-A amplifier.

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

Generalizations

Multivariate functions

Even symmetry:

A function $f:\mathbb {R} ^{n}\to \mathbb {R}$ is calledeven symmetric if:

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}

Odd symmetry:

A function $f:\mathbb {R} ^{n}\to \mathbb {R}$ is calledodd symmetric if:

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}

Complex-valued functions

The definitions for even and odd symmetry forcomplex-valued functions of a real argument are similar to the real case. Insignal processing, a similar symmetry is sometimes considered, which involvescomplex conjugation.^[5]^[6]

Conjugate symmetry:

A complex-valued function of a real argument $f:\mathbb {R} \to \mathbb {C}$ is calledconjugate symmetric if

f(x)={\overline {f(-x)}}\quad {\text{for all }}x\in \mathbb {R}

A complex valued function is conjugate symmetric if and only if itsreal part is an even function and itsimaginary part is an odd function.

A typical example of a conjugate symmetric function is thecis function

x\to e^{ix}=\cos x+i\sin x

Conjugate antisymmetry:

A complex-valued function of a real argument $f:\mathbb {R} \to \mathbb {C}$ is calledconjugate antisymmetric if:

f(x)=-{\overline {f(-x)}}\quad {\text{for all }}x\in \mathbb {R}

A complex valued function is conjugate antisymmetric if and only if itsreal part is an odd function and itsimaginary part is an even function.

Finite length sequences

The definitions of odd and even symmetry are extended toN-point sequences (i.e. functions of the form $f:\left\{0,1,\ldots ,N-1\right\}\to \mathbb {R}$ ) as follows:^[6]^{: p. 411}

Even symmetry:

AN-point sequence is calledconjugate symmetric if

f(n)=f(N-n)\quad {\text{for all }}n\in \left\{1,\ldots ,N-1\right\}.

Such a sequence is often called apalindromic sequence; see alsoPalindromic polynomial.

Odd symmetry:

AN-point sequence is calledconjugate antisymmetric if

f(n)=-f(N-n)\quad {\text{for all }}n\in \left\{1,\ldots ,N-1\right\}.

Such a sequence is sometimes called ananti-palindromic sequence; see alsoAntipalindromic polynomial.

See also

Hermitian function for a generalization in complex numbers
Taylor series
Fourier series
Holstein–Herring method
Parity (physics)

Notes

^^a ^bGel'Fand, I. M.;Glagoleva, E. G.; Shnol, E. E. (1990).Functions and Graphs. Birkhäuser.ISBN 0-8176-3532-7.
^W., Weisstein, Eric."Odd Function".mathworld.wolfram.com.{{cite web}}: CS1 maint: multiple names: authors list (link)
^W., Weisstein, Eric."Even Function".mathworld.wolfram.com.{{cite web}}: CS1 maint: multiple names: authors list (link)
^Berners, Dave (October 2005)."Ask the Doctors: Tube vs. Solid-State Harmonics".UA WebZine. Universal Audio. Retrieved2016-09-22.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.
^Oppenheim, Alan V.;Schafer, Ronald W.; Buck, John R. (1999).Discrete-time signal processing (2nd ed.). Upper Saddle River, N.J.: Prentice Hall. p. 55.ISBN 0-13-754920-2.
^^a ^bProakis, John G.; Manolakis, Dimitri G. (1996),Digital Signal Processing: Principles, Algorithms and Applications (3 ed.), Upper Saddle River, NJ: Prentice-Hall International,ISBN 9780133942897, sAcfAQAAIAAJ

References

Gelfand, I. M.; Glagoleva, E. G.; Shnol, E. E. (2002) [1969],Functions and Graphs, Mineola, N.Y: Dover Publications

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