In
mathematics, an even function is a
real function 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 y-axis, 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.
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 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 functions
A
real functionf is even if, for every x in its domain, −x is also in its domain and[1]: p. 11
or equivalently
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.
A real function f is odd if, for every x in its domain, −x is also in its domain and[1]: p. 72
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 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 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 and the odd part of the function, and are defined by
and
It is straightforward to verify that is even, is odd, and
This decomposition is unique since, if
where g is even and h 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 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 are considered, and these concepts are thus supposed to be defined for the considered functions.
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. , the result of the integral over that interval is zero; that is[2]
.
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
.
Series
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 . 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;
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;
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.
Generalizations
Multivariate functions
Even symmetry:
A function is called even symmetric if:
Odd symmetry:
A function is called odd symmetric if:
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, a similar symmetry is sometimes considered, which involves
complex conjugation.[4][5]
Conjugate symmetry:
A complex-valued function of a real argument is called conjugate symmetric if
A complex valued function is conjugate symmetric if and only if its
real part is an even function and its
imaginary part is an odd function.
A typical example of a conjugate symmetric function is the
cis function
Conjugate antisymmetry:
A complex-valued function of a real argument is called conjugate antisymmetric if:
A complex valued function is conjugate antisymmetric if and only if its
real part is an odd function and its
imaginary part 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 ) as follows:[5]: p. 411
Even symmetry:
A N-point sequence is called conjugate symmetric if
Such a sequence is often called a palindromic sequence; see also
Palindromic polynomial.
Odd symmetry:
A N-point sequence is called conjugate antisymmetric if
^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.