Even and odd functions

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

Not to be confused with Even and odd numbers.

They are named for the parity of the powers of the power functions which satisfy each condition: the function $f(x)=x^{n}$ 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.

The $x\mapsto |x|,$

absolute value

$x\mapsto x^{2},$

$x\mapsto x^{4},$

$\cos ,$

cosine

$\cosh ,$

hyperbolic cosine

$x\mapsto \exp(-x^{2}).$

Gaussian function

If a function is both even and odd, it is equal to 0 everywhere it is defined.

If a function is odd, the of that function is an even function.

absolute value

graded algebra

The of an even function is odd.

derivative

The derivative of an odd function is even.

The 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. $[-A,A]$ , the result of the integral over that interval is zero; that is^[2]
$\int _{-A}^{A}f(x)\,dx=0$ .

integral

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
$\int _{-A}^{A}f(x)\,dx=2\int _{0}^{A}f(x)\,dx$ .

fundamental

symmetric

class-A amplifier

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 $V_{\text{out}}(t)=f(V_{\text{in}}(t))$ . The type of harmonics produced depend on the response function f:^[3]

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[edit]

Multivariate functions[edit]

Even symmetry:

A function $f:\mathbb {R} ^{n}\to \mathbb {R}$ is called even symmetric if:

for a generalization in complex numbers

Hermitian function

Taylor series

Fourier series

Holstein–Herring method

Parity (physics)

; Glagoleva, E. G.; Shnol, E. E. (2002) [1969], Functions and Graphs, Mineola, N.Y: Dover Publications

Even and odd functions

absolute value

cosine

hyperbolic cosine

Gaussian function

absolute value

graded algebra

derivative

integral

fundamental

symmetric

class-A amplifier

Generalizations[edit]

Multivariate functions[edit]

Hermitian function

Taylor series

Fourier series

Holstein–Herring method

Parity (physics)

Gelfand, I. M.