Even and odd functions

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

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

Even and odd functions are two classes of functions in mathematics that have certain symmetry properties:

• a real function is even if and only if its function graph is axisymmetric to the y -axis , and
• odd if its function graph is point-symmetric to the coordinate origin .

In school mathematics , the examination of a functional diagram for these symmetries is one of the first steps in a curve discussion .

definition

A real function with a definition set symmetrical with respect to zero is called even if for all arguments ${\displaystyle f\colon D\to \mathbb {R} }$ ${\displaystyle D\subseteq \mathbb {R} }$ ${\displaystyle x\in D}$

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

holds, and it is called odd if for all ${\displaystyle x\in D}$

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

applies. A real function is clearly even if its function graph is axisymmetric to the y -axis , and odd if its function graph is point-symmetric to the coordinate origin .

Examples

Even functions

• the constant function ${\displaystyle f(x)=1}$
• the amount function ${\displaystyle f(x)=|x|}$
• the normal parabola ${\displaystyle f(x)=x^{2}}$
• the cosine function ${\displaystyle f(x)=\cos(x)}$
• the secant function ${\displaystyle f(x)=\sec(x)}$
• the Gaussian bell curve ${\displaystyle f(x)=\exp(-x^{2}/2)}$

Odd functions

• the sign function ${\displaystyle f(x)=\operatorname {sgn} (x)}$
• the identical function ${\displaystyle f(x)=x}$
• the cubic function ${\displaystyle f(x)=x^{3}}$
• the sine function ${\displaystyle f(x)=\sin(x)}$
• the tangent function ${\displaystyle f(x)=\tan(x)}$
• the Gaussian error function ${\displaystyle f(x)=\operatorname {erf} (x)}$

The only function that is even and odd at the same time is the null function . ${\displaystyle f(x)=0}$

More general examples

• A power function is even if and only if the exponent is even and odd if and only if the exponent is odd.
  ${\displaystyle f(x)=ax^{n}}$
  ${\displaystyle a\neq 0}$ ${\displaystyle n}$ ${\displaystyle n}$
• A polynomial function is even if and only if all odd-numbered coefficients are equal to zero, and odd if and only if all even-numbered coefficients are equal to zero.
  ${\displaystyle f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\dotsb +a_{n}x^{n}}$
  ${\displaystyle a_{1},a_{3},a_{5},\dotsc }$ ${\displaystyle a_{0},a_{2},a_{4},\dotsc }$
• A trigonometric polynomial is even if and only if all are coefficients and odd if and only if all coefficients are.
  ${\displaystyle a_{0}+a_{1}\cos(x)+b_{1}\sin(x)+\dotsb +a_{n}\cos(nx)+b_{n}\sin(nx)}$
  ${\displaystyle b_{i}=0}$ ${\displaystyle a_{i}=0}$

Disassembly

There are also functions that are neither even nor odd, for example the function . However, every function with a definition set that is symmetrical with respect to zero can be written as the sum of an even and an odd function. This means ${\displaystyle f(x)=x+1}$ ${\displaystyle D\subseteq \mathbb {R} }$

${\displaystyle f(x)=f_{\text{g}}(x)+f_{\text{u}}(x)}$,

in which

${\displaystyle f_{\text{g}}(x)={\frac {f(x)+f(-x)}{2}}}$

the even part of the function and

${\displaystyle f_{\text{u}}(x)={\frac {f(x)-f(-x)}{2}}}$

represents the odd part of the function. This decomposition of a function into even and odd components is unambiguous; H. there is no other way to break a function down into odd and even components. This follows from the facts that the sets of all even / odd functions form a subspace of the space of all functions and the only function that is both even and odd is the null function.

properties

Algebraic properties

• Every multiple of an even or odd function is again even or odd.
• The sum of two even functions is again even.
• The sum of two odd functions is again odd.
• The product of two even functions is again even.
• The product of two odd functions is even.
• The product of an even and an odd function is odd.
• The quotient of two even functions is again even.
• The quotient of two odd functions is even.
• The quotient of an even and an odd function is odd.
• The composition of any function with an even function is even.
• The composition of an odd function with an odd function is odd.

Analytical properties

• In the zero point (if this is included in the definition range) every odd function has the function value zero.
• The derivative of an even differentiable function is odd, the derivative of an odd differentiable function is even.
• The definite integral of an odd continuous function results when the integration limits are symmetrical around the zero point. ${\displaystyle 0}$
• The Taylor series with the expansion point of an even (odd) function contains only even (odd) powers. ${\displaystyle x=0}$
• The Fourier series of an even (odd) function contains only cosine (sine) terms.

Generalizations

More generally defined in the algebra by the above definition also even and odd functions between two quantities and on which a link with additive inverse is given, for example (additive) groups , rings , body or vector spaces . In this way, for example, even and odd complex functions or even and odd vector-valued functions can be defined. ${\displaystyle f\colon X\to Y}$ ${\displaystyle X}$ ${\displaystyle Y}$

In mathematical physics , the concept of even and odd functions is generalized by the concept of parity . This is especially important for wave functions in quantum mechanics, for example .

literature

• Marc Hensel: Curve discussion . Learning and exercise book for the mathematics Abitur examination. 1st edition. Books on Demand , Norderstedt 2010, ISBN 978-3-8391-4025-3 .
• Harro Heuser : Textbook of Analysis . 8th edition. Part 1. BG Teubner, Stuttgart 1988, ISBN 3-519-12231-6 .

Web links

• Eric W. Weisstein : Even Function . In: MathWorld (English).
• Eric W. Weisstein : Odd Function . In: MathWorld (English).
• yark, matte, Cam McLeman: Even and odd functions . In: PlanetMath . (English)
• Simple explanation of the curve discussion with symmetry investigation for a rational function
• Exemplary curve discussion at MathePlanet