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 inverses. 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. 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
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. 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'':
:
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
*
*
*
cosine
*
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 ...
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), ''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 of 180 degrees about the origin.
Examples of odd functions are:
*The identity function
*
* sine
* hyperbolic sine
*The error function
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 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 is even, is odd, and
:
Conversely, if
:
where is even and is odd, then and since
:
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
:.
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 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
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, 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, Taylor series, 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.