In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions 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.
The sine function and all of its Taylor polynomials are odd functions. This image shows $\sin(x)$ and its Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13.
The cosine function and all of its Taylor polynomials are even functions. This image shows $\cos(x)$ and its Taylor approximation of degree 4.
Definition and examplesEdit
Evenness and oddness are generally considered for real functions, that is real-valued functions of a real variable. However, the concepts may be more generally defined for functions whose domain and codomain both have a notion of additive inverse. This includes abelian groups, all rings, all fields, and all vector spaces. Thus, for example, a real function could be odd or even (or neither), as could a complex-valued function of a vector variable, and so on.
The given examples are real functions, to illustrate the symmetry of their graphs.
Even functionsEdit
$f(x)=x^{2}$ 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 are in the domain of f:^{[1]}^{: p. 11 }
$f(x)=f(-x)$
(Eq.1)
or equivalently if the following equation holds for all such x:
$f(x)-f(-x)=0.$
Geometrically, the graph of an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis.
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:^{[1]}^{: p. 72 }
$-f(x)=f(-x)$
(Eq.2)
or equivalently if the following equation holds for all such x:
$f(x)+f(-x)=0.$
Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin.
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 decompositionEdit
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
$f_{\text{e}}(x)={\frac {f(x)+f(-x)}{2}}$
(Eq.3)
and
$f_{\text{o}}(x)={\frac {f(x)-f(-x)}{2}}$
(Eq.4)
then $f_{\text{e}}$ is even, $f_{\text{o}}$ is odd, and
$f(x)=f_{\text{e}}(x)+f_{\text{o}}(x).$
Conversely, if
$f(x)=g(x)+h(x),$
where g is even and h is odd, then $g=f_{\text{e}}$ and $h=f_{\text{o}},$ since
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
Any linear combination of even functions is even, and the even functions form a vector space over the reals. 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 the 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 commutative algebra over the reals. However, the odd functions do not form an algebra over the reals, as they are not closed under multiplication.
In the following, properties involving derivatives, Fourier series, Taylor series, and so on suppose that these concepts are defined of the functions that are considered.
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). For an odd function that is integrable over a symmetric interval, e.g. $[-A,A]$, the result of the integral over that interval is zero; that is^{[2]}
$\int _{-A}^{A}f(x)\,dx=0$.
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
$\int _{-A}^{A}f(x)\,dx=2\int _{0}^{A}f(x)\,dx$.
SeriesEdit
The Maclaurin series of an even function includes only even powers.
The Maclaurin series of an odd function includes only odd powers.
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 $V_{\text{out}}(t)=f(V_{\text{in}}(t))$. The type of harmonics produced depend on the response function f:^{[3]}
When the response function is even, the resulting signal will consist of only even harmonics of the input sine wave; $0f,2f,4f,6f,\dots$
The fundamental is also an odd harmonic, so will not be present.
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 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.
GeneralizationsEdit
Multivariate functionsEdit
Even symmetry:
A function $f:\mathbb {R} ^{n}\to \mathbb {R}$ is called even 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 called odd 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 functionsEdit
The definitions for even and odd symmetry for 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 $f:\mathbb {R} \to \mathbb {C}$ is called even symmetric if:
$f(x)={\overline {f(-x)}}\quad {\text{for all }}x\in \mathbb {R}$
Odd symmetry:
A complex-valued function of a real argument $f:\mathbb {R} \to \mathbb {C}$ is called odd symmetric if:
$f(x)=-{\overline {f(-x)}}\quad {\text{for all }}x\in \mathbb {R}$
Finite length sequencesEdit
The definitions of odd and even symmetry are extended to N-point sequences (i.e. functions of the form $f:\left\{0,1,\ldots ,N-1\right\}\to \mathbb {R}$) as follows:^{[4]}^{: p. 411 }
Even symmetry:
A N-point sequence is called even 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 a palindromic sequence; see also Palindromic polynomial.
Odd symmetry:
A N-point sequence is called odd symmetric if
$f(n)=-f(N-n)\quad {\text{for all }}n\in \left\{1,\ldots ,N-1\right\}.$
^Berners, Dave (October 2005). "Ask the Doctors: Tube vs. Solid-State Harmonics". UA WebZine. Universal Audio. Retrieved 2016-09-22. 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.
^Proakis, 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
ReferencesEdit
Gelfand, I. M.; Glagoleva, E. G.; Shnol, E. E. (2002) [1969], Functions and Graphs, Mineola, N.Y: Dover Publications