Even functions

${\displaystyle f(x)=x^{2}}$ is an example of an even function.

Let f be a real-valued function of a real variable. Then f is even if the following equation holds for all x such that x and −x are in the domain of f:^{[1]: p. 11}

${\displaystyle f(-x)=f(x)}$

(Eq.1)

or equivalently if the following equation holds for all such x:

${\displaystyle f(x)-f(-x)=0.}$

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 ${\displaystyle x\mapsto |x|,}$
• ${\displaystyle x\mapsto x^{2},}$
• ${\displaystyle x\mapsto x^{4},}$
• cosine ${\displaystyle \cos ,}$
• hyperbolic cosine ${\displaystyle \cosh ,}$
• Gaussian function ${\displaystyle x\mapsto \exp(-x^{2}).}$

Odd functions

${\displaystyle f(x)=x^{3}}$ is an example of an odd function.

Again, let f be a real-valued function of a real variable. Then f is odd if the following equation holds for all x such that x and −x are in the domain of f:^{[1]: p. 72}

${\displaystyle f(-x)=-f(x)}$

(Eq.2)

or equivalently if the following equation holds for all such x:

${\displaystyle f(x)+f(-x)=0.}$

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:

• The sign function ${\displaystyle x\mapsto \operatorname {sgn}(x),}$
• The identity function ${\displaystyle x\mapsto x,}$
• ${\displaystyle x\mapsto x^{3},}$
• sine ${\displaystyle \sin ,}$
• hyperbolic sine ${\displaystyle \sinh ,}$
• The error function ${\displaystyle \operatorname {erf} .}$

${\displaystyle f(x)=x^{3}+1}$ is neither even nor odd.

Uniqueness

• If a function is both even and odd, it is equal to 0 everywhere it is defined.
• If a function is odd, the absolute value of that function is an even function.

Addition and subtraction

• The sum of two even functions is even.
• The sum of two odd functions is odd.
• The difference between two odd functions is odd.
• The difference between two even functions is even.
• The sum of an even and odd function is not even or odd, unless one of the functions is equal to zero over the given domain.

Multiplication and division

• The product of two even functions is an even function.
  • That implies that product of any number of even functions is an even function as well.
• The product of two odd functions is an even function.
• The product of an even function and an odd function is an odd function.
• The quotient of two even functions is an even function.
• The quotient of two odd functions is an even function.
• The quotient of an even function and an odd function is an odd function.

Composition

• The composition of two even functions is even.
• The composition of two odd functions is odd.
• 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).

Basic analytic properties

• The derivative of an even function is odd.
• The derivative of an odd function is even.
• 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. ${\displaystyle [-A,A]}$, the result of the integral over that interval is zero; that is^[2]

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

• 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

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

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 Fourier analysis § Symmetry properties)
• The Fourier transform of a purely real-valued odd function is imaginary and odd. (see Fourier analysis § Symmetry properties)

Multivariate functions

Even symmetry:

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

${\displaystyle f(x_{1},x_{2},\ldots ,x_{n})=f(-x_{1},-x_{2},\ldots ,-x_{n})\quad {\text{for all }}x_{1},\ldots ,x_{n}\in \mathbb {R} }$

Odd symmetry:

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

${\displaystyle f(x_{1},x_{2},\ldots ,x_{n})=-f(-x_{1},-x_{2},\ldots ,-x_{n})\quad {\text{for all }}x_{1},\ldots ,x_{n}\in \mathbb {R} }$

Complex-valued functions

The definitions for even and odd symmetry for complex-valued functions of a real argument are similar to the real case but involve complex conjugation.

Even symmetry:

A complex-valued function of a real argument ${\displaystyle f:\mathbb {R} \to \mathbb {C} }$ is called even symmetric if:

${\displaystyle f(x)={\overline {f(-x)}}\quad {\text{for all }}x\in \mathbb {R} }$

Odd symmetry:

A complex-valued function of a real argument ${\displaystyle f:\mathbb {R} \to \mathbb {C} }$ is called odd symmetric if:

${\displaystyle f(x)=-{\overline {f(-x)}}\quad {\text{for all }}x\in \mathbb {R} }$

Finite length sequences

The definitions of odd and even symmetry are extended to N-point sequences (i.e. functions of the form ${\displaystyle f:\left\{0,1,\ldots ,N-1\right\}\to \mathbb {R} }$) as follows:^{[4]: p. 411}

Even symmetry:

A N-point sequence is called even symmetric if

${\displaystyle f(n)=f(N-n)\quad {\text{for all }}n\in \left\{1,\ldots ,N-1\right\}.}$

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

Odd symmetry:

A N-point sequence is called odd symmetric if

${\displaystyle f(n)=-f(N-n)\quad {\text{for all }}n\in \left\{1,\ldots ,N-1\right\}.}$

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