Even and odd functions

thumb|The [[sine function and all of its Taylor polynomials are odd functions.]]

thumb|The [[cosine function and all of its Taylor polynomials are even functions.]]

In mathematics, an even function is a real function such that <math>f(-x)=f(x)</math> for every <math>x</math> in its domain. Similarly, an odd function is a function such that <math>f(-x)=-f(x)</math> for every <math>x</math> in its domain.

They are named for the parity of the powers of the power functions which satisfy each condition: the function <math>f(x) = x^n</math> 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 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.

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

right|thumb|<math>f(x)=x^2</math> is an example of an even function.

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:

The absolute value <math>x \mapsto |x|,</math>
<math>x \mapsto x^2,</math>
<math>x \mapsto x^n</math> for any even integer <math>n,</math>
cosine <math>\cos,</math>
hyperbolic cosine <math>\cosh,</math>
Gaussian function <math>x \mapsto \exp (-x^2). </math>

Odd functions

right|thumb|<math>f(x)=x^3</math> is an example of an odd function.

A real function is odd if, for every in its domain, is also in its domain and

:<math>\int_{-A}^{A} f(x)\,dx = 0</math>.
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 <math>[-A,A]</math>, the integral over that interval is twice the integral from 0 to A; that is
:<math>\int_{-A}^{A} f(x)\,dx = 2\int_{0}^{A} f(x)\,dx</math>.
This property is also true for the improper integral when <math>A = \infty</math>, provided the integral from 0 to <math>\infty</math> 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 of a periodic even function includes only cosine terms.
The Fourier series of a periodic odd function includes only sine terms.
The Fourier transform 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, 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 <math>V_\text{out}(t) = f(V_\text{in}(t))</math>. The type of harmonics 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; <math>0f, 2f, 4f, 6f, \dots </math>
The fundamental is also an odd harmonic, so will not be present.
A simple example is a full-wave rectifier.
The <math>0f</math> 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; <math>1f, 3f, 5f, \dots </math>
The output signal will be half-wave symmetric.
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; <math>1f, 2f, 3f, \dots </math>
Simple examples are a half-wave rectifier, and clipping in an asymmetrical class-A amplifier.

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 <math>f: \mathbb{R}^n \to \mathbb{R} </math> is called even symmetric if:

:<math>f(x_1,x_2,\ldots,x_n)=f(-x_1,-x_2,\ldots,-x_n) \quad \text{for all } x_1,\ldots,x_n \in \mathbb{R}</math>

Odd symmetry:

A function <math>f: \mathbb{R}^n \to \mathbb{R} </math> is called odd symmetric if:

:<math>f(x_1,x_2,\ldots,x_n)=-f(-x_1,-x_2,\ldots,-x_n) \quad \text{for all } x_1,\ldots,x_n \in \mathbb{R}</math>

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.

Even symmetry:

A N-point sequence is called conjugate symmetric if

:<math>f(n) = f(N-n) \quad \text{for all } n \in \left\{ 1,\ldots,N-1 \right\}.</math>

Such a sequence is often called a palindromic sequence; see also Palindromic polynomial.

Odd symmetry:

A N-point sequence is called conjugate antisymmetric if

:<math>f(n) = -f(N-n) \quad \text{for all } n \in \left\{1,\ldots,N-1\right\}. </math>

Such a sequence is sometimes called an anti-palindromic sequence; see also Antipalindromic polynomial.