Even and odd functions

They are named for the parity of the powers of the power functions which satisfy each condition: the function

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.

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.

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 f is odd if, for every x in its domain, −x is also in its domain and[1]: p. 72

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.

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

This decomposition is unique since, if where g is even and h is odd, then

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 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).

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.

The type of harmonics produced depend on the response function f:[3] 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.

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.

[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.

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.