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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, even functions and odd functions are functions which satisfy particular
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
relations, with respect to taking
additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (op ...
s. They are important in many areas of
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
, especially the theory of
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
and
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
. They are named for the
parity Parity may refer to: * Parity (computing) ** Parity bit in computing, sets the parity of data for the purpose of error detection ** Parity flag in computing, indicates if the number of set bits is odd or even in the binary representation of the ...
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 An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
, and it is an odd function if ''n'' is an odd integer.


Definition and examples

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 Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
and codomain both have a notion of
additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (op ...
. This includes
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s, all rings, all fields, and all
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s. 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 Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
of their graphs.


Even functions

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'': 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. Examples of even functions are: *The
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
x \mapsto , x, , *x \mapsto x^2, *x \mapsto x^4, * cosine \cos, *
hyperbolic cosine In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...
\cosh.


Odd functions

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'': 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 Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
, meaning that its graph remains unchanged after
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
of 180 degrees about the origin. Examples of odd functions are: *The identity function x \mapsto x, *x \mapsto x^3, *
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
\sin, * hyperbolic sine \sinh, *The
error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non- elementa ...
\operatorname.


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, the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
of that function is an even function.


Addition and subtraction

* The sum of two even functions is even. * The sum of two odd functions is odd. * The difference 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 given
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
.


Multiplication and division

* The product of two even functions is an even function. ** That implies that product of any number of even functions is an even function as well. * 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 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.


Composition

* The 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 any function with an even function is even (but not vice versa).


Even–odd decomposition

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 and then f_\text is even, f_\text is odd, and : f(x)=f_\text(x) + f_\text(x). Conversely, if :f(x)=g(x)+h(x), where is even and is odd, then g=f_\text and h=f_\text, since : \begin 2f_\text(x) &=f(x)+f(-x)= g(x) + g(-x) +h(x) +h(-x) = 2g(x),\\ 2f_\text(x) &=f(x)-f(-x)= g(x) - g(-x) +h(x) -h(-x) = 2h(x). \end For example, the
hyperbolic cosine In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...
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 :e^x=\underbrace_ + \underbrace_.


Further algebraic properties

* Any linear combination of even functions is even, and the even functions form a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
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 The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
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 In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
over the real numbers by this property, as well as some of those above. *The even functions form a
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
over the reals. However, the odd functions do ''not'' form an algebra over the reals, as they are not
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
under multiplication.


Analytic properties

A function's being odd or even does not imply
differentiability In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
, or even continuity. For example, the Dirichlet function is even, but is nowhere continuous. In the following, properties involving
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s,
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
,
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
, and so on suppose that these concepts are defined of the functions that are considered.


Basic analytic properties

* The
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of an even function is odd. * The derivative of an odd function is even. * The
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
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/math>, the result of the integral over that interval is zero; that is *:\int_^ 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_^ f(x)\,dx = 2\int_^ f(x)\,dx.


Series

* The
Maclaurin series Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin ...
of an even function includes only even powers. * The Maclaurin series of an odd function includes only odd powers. * The
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
of a periodic even function includes only cosine terms. * The Fourier series of a periodic odd function includes only
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
terms. *The
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
of a purely real-valued even function is real and even. (see ) *The Fourier transform of a purely real-valued odd function is imaginary and odd. (see )


Harmonics

In
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
, harmonic distortion occurs when a
sine wave A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...
signal is sent through a memory-less
nonlinear system In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
, 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(t) = f(V_\text(t)). The type of
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...
s produced depend on the response function ''f'': * 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. ** A simple example is a full-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-wave symmetric. ** A simple example is clipping in a symmetric push-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 asymmetrical
class-A amplifier In electronics, power amplifier classes are letter symbols applied to different power amplifier types. The class gives a broad indication of an amplifier's characteristics and performance. The classes are related to the time period that the active ...
. Note that this does not hold true for more complex waveforms. A
sawtooth wave The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle. A single sawtooth, or an intermittently triggered sawtooth, is called ...
contains both even and odd harmonics, for instance. After even-symmetric full-wave rectification, it becomes a
triangle wave A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function. Like a square wave, the triangle wave contains only odd harmonics. However, ...
, which, other than the DC offset, contains only odd harmonics.


Generalizations


Multivariate functions

Even symmetry: A function f: \mathbb^n \to \mathbb is called ''even symmetric'' if: :f(x_1,x_2,\ldots,x_n)=f(-x_1,-x_2,\ldots,-x_n) \quad \text x_1,\ldots,x_n \in \mathbb Odd symmetry: A function f: \mathbb^n \to \mathbb is called ''odd symmetric'' if: :f(x_1,x_2,\ldots,x_n)=-f(-x_1,-x_2,\ldots,-x_n) \quad \text x_1,\ldots,x_n \in \mathbb


Complex-valued functions

The definitions for even and odd symmetry for
complex-valued In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
functions of a real argument are similar to the real case but involve
complex conjugation In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
. Even symmetry: A complex-valued function of a real argument f: \mathbb \to \mathbb is called ''even symmetric'' if: :f(x)=\overline \quad \text x \in \mathbb Odd symmetry: A complex-valued function of a real argument f: \mathbb \to \mathbb is called ''odd symmetric'' if: :f(x)=-\overline \quad \text x \in \mathbb


Finite length sequences

The definitions of odd and even symmetry are extended to ''N''-point sequences (i.e. functions of the form f: \left\ \to \mathbb) as follows: Even symmetry: A ''N''-point sequence is called ''even symmetric'' if :f(n) = f(N-n) \quad \text n \in \left\. 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 n \in \left\. Such a sequence is sometimes called an anti-palindromic sequence; see also Antipalindromic polynomial.


See also

* Hermitian function for a generalization in complex numbers *
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
*
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
* Holstein–Herring method *
Parity (physics) In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point ref ...


Notes


References

*{{Citation , last=Gelfand , first=I. M. , last2=Glagoleva , first2=E. G. , last3=Shnol , first3=E. E. , author-link=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 Calculus Parity (mathematics) Types of functions