
In
mathematics, even functions and odd functions are
functions which satisfy particular
symmetry 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 (opp ...
s. They are important in many areas of
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, 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 r ...
of the powers of the
power functions which satisfy each condition: the function
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 language ...
, and it is an odd function if ''n'' is an odd integer.
Definition and examples
Evenness and oddness are generally considered for
real function
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an inter ...
s, 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
*Do ...
and
codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...
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 (opp ...
. 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 com ...
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 ...
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 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'':
:
Geometrically, the graph of an even function is
symmetric
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 definit ...
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
*
*
*
cosine
*
hyperbolic cosine
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'':
:
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), a Wolverine comic book mini-series published by Marvel Comics in 2002
* ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...
, 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 degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathemati ...
s about the origin.
Examples of odd functions are:
*The identity function
*
*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 ...
* hyperbolic sine
*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 ...
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 of that function is an even function.
Addition and subtraction
* The sum
Sum most commonly means the total of two or more numbers added together; see addition.
Sum can also refer to:
Mathematics
* Sum (category theory), the generic concept of summation in mathematics
* Sum, the result of summation, the additio ...
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
*Do ...
.
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
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
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 is even, is odd, and
:
Conversely, if
:
where is even and is odd, then and 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
:.
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 ...
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 mo ...
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. Promi ...
over the reals. However, the odd functions do ''not'' form an algebra over the reals, as they are not closed 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 se ...
, 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.
Series
* The Maclaurin series 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
In signal processing, distortion is the alteration of the original shape (or other characteristic) of a signal. In communications and electronics it means the alteration of the waveform of an information-bearing signal, such as an audio signal ...
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 other ...
, 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'', ...
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
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 definit ...
.
** 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.
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 a ...
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
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 for ...
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 \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 se ...
*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 refle ...
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