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 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.
The concept of even and odd functions appears to date back to the early 18th century, with Leonhard 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 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.
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.
A real function 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 with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis.
Examples of even functions are:
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 of 180 degrees about the origin.
If is in the domain of an odd function , then .
Examples of odd functions are:
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
and
It is straightforward to verify that is even, is odd, and
This decomposition is unique since, 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
Fourier's sine and cosine transforms also perform evenâÂÂodd 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).
A function's being odd or even does not imply differentiability, or continuity. For example, the Dirichlet function is even, but is nowhere continuous.
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.
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:
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.
Even symmetry:
A function is called even symmetric if:
Odd symmetry:
A function is called odd symmetric if:
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.
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.
The definitions of odd and even symmetry are extended to N-point sequences (i.e. functions of the form ) as follows:
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
Such a sequence is sometimes called an anti-palindromic sequence; see also Antipalindromic polynomial.