
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an even function is a
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 ...
such that
for every
in its
domain. Similarly, an odd function is a function such that
for every
in its domain.
They are named for the
parity of the powers of the
power functions which satisfy each condition: the function
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 is
self-symmetric with respect to the and odd functions are those whose graph is self-symmetric with respect to the
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
Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
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
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 and
codomain
In mathematics, a codomain, counter-domain, or set of destination of a function is a 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 ...
both have a notion of
additive inverse
In mathematics, the additive inverse of an element , denoted , is the element that when added to , yields the additive identity, 0 (zero). In the most familiar cases, this is the number 0, but it can also refer to a more generalized zero el ...
. 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 commu ...
s, all
rings, all
fields, and all
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s. Thus, for example, a real function could be odd or even (or neither), as could a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued function of a vector variable, and so on.
The given examples are real functions, to illustrate the
symmetry
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
of their
graphs.
Even functions
A
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 ...
is even if, for every in its domain, is also in its domain and
or equivalently
Geometrically, the graph of an even function is
symmetric
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
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 x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
*
*
for any even integer
*
cosine
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 opposite that ...
*
hyperbolic cosine
*
Gaussian function
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form
f(x) = \exp (-x^2)
and with parametric extension
f(x) = a \exp\left( -\frac \right)
for arbitrary real number, rea ...
Odd functions
A real function is odd if, for every in its domain, is also in its domain and
[
or equivalently
Geometrically, the graph of an odd function has rotational symmetry with respect to the origin, meaning that its graph remains unchanged after ]rotation
Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
of 180 degrees about the origin.
If is in the domain of an odd function , then .
Examples of odd functions are:
*The sign function
In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is a function that has the value , or according to whether the sign of a given real number is positive or negative, or the given number is itself zer ...
*The identity function
* for any odd integer
*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 opposite th ...
\sin,
*hyperbolic sine
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 unit circle, circle with a unit radius, the points form the right ha ...
\sinh,
*The error function
In mathematics, the error function (also called the Gauss error function), often denoted by , is a function \mathrm: \mathbb \to \mathbb defined as:
\operatorname z = \frac\int_0^z e^\,\mathrm dt.
The integral here is a complex Contour integrat ...
\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 x 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.
Multiplication and division
* The product and quotient 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 the reciprocal 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
* The composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
* Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...
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 is invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
, 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 the even part (or the even component) and the odd part (or the odd component) of the function, and are defined by
f_\text(x) = \frac ,
and
f_\text(x) = \frac .
It is straightforward to verify that f_\text is even, f_\text is odd, and f=f_\text+f_\text.
This decomposition is unique since, 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 and the hyperbolic sine
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 unit circle, circle with a unit radius, the points form the right ha ...
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_.
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
* Any linear combination
In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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. As an example, the direct sum of two abelian groups A and B is anothe ...
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 . The index set is usually the set of nonnegative integers or the set of integers, ...
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 ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
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, or even continuity. For example, the Dirichlet function
In mathematics, the Dirichlet function is the indicator function \mathbf_\Q of the set of rational numbers \Q, i.e. \mathbf_\Q(x) = 1 if is a rational number and \mathbf_\Q(x) = 0 if is not a rational number (i.e. is an irrational number).
\mathb ...
is even, but is nowhere continuous.
In the following, properties involving derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s, Fourier series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
, 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 ...
are considered, and these concepts are thus supposed to be defined for the considered functions.
Basic analytic properties
* The derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
of an even function is odd.
* The derivative of an odd function is even.
* If an odd function is integrable over a bounded symmetric interval A,A/math>, the integral over that interval is zero; that is
*:\int_^ f(x)\,dx = 0.
** This implies that the Cauchy 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/math>, the integral over that interval is twice the integral from 0 to ''A''; that is
*:\int_^ f(x)\,dx = 2\int_^ f(x)\,dx.
** This property is also true for the improper integral
In mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals (or, equivalently, Darboux integral ...
when A = \infty, provided the integral from 0 to \infty converges.
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 an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
of a periodic even function includes only cosine
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 opposite that ...
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 opposite th ...
terms.
*The Fourier transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
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 audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
, harmonic distortion occurs when a sine wave
A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodic function, periodic wave whose waveform (shape) is the trigonometric function, trigonometric sine, sine function. In mechanics, as a linear motion over time, this is ''simple ...
signal is sent through a memory-less nonlinear system
In mathematics and science, a nonlinear system (or a non-linear 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, mathem ...
, 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
In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st har ...
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 () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
.
** 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 Amplifier#Power_amplifiers, power amplifier types. The class gives a broad indication of an amplifier's Electrical efficiency, efficiency, linearity and other character ...
.
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.
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 functions of a real argument are similar to the real case. In signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
, a similar symmetry is sometimes considered, which involves 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 numbers, then the complex conjugate of a + bi is a - ...
.[
][
Conjugate symmetry:
A complex-valued function of a real argument f: \mathbb \to \mathbb is called ''conjugate symmetric'' if
:f(x)=\overline \quad \text x \in \mathbb
A complex valued function is conjugate symmetric if and only if its ]real part
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 ...
is an even function and its imaginary part
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 ...
is an odd function.
A typical example of a conjugate symmetric function is the cis function
:x \to e^=\cos x + i\sin x
Conjugate antisymmetry:
A complex-valued function of a real argument f: \mathbb \to \mathbb is called ''conjugate antisymmetric'' if:
:f(x)=-\overline \quad \text x \in \mathbb
A complex valued function is conjugate antisymmetric if and only if its real part
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 ...
is an odd function and its imaginary part
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 ...
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 f: \left\ \to \mathbb) as follows:
Even symmetry:
A ''N''-point sequence is called ''conjugate 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 ''conjugate antisymmetric'' 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 an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
* 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 , last1=Gelfand , first1=I. M. , last2=Glagoleva , first2=E. G. , last3=Shnol , first3=E. E. , author-link1=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