Even and odd functions

Even and odd functions in mathematics two classes of functions that possess certain symmetry properties. A real function is exactly then just when you graph function is axially symmetric to the y- axis and odd if their function graph is point symmetric to the origin. In school mathematics, the study of a functional diagram of these symmetries is one of the first steps toward a curve discussion.

  • 4.1 Algebraic properties
  • 4.2 Analytical properties

Definition

A real function with respect to the zero symmetric domain is called even if for all arguments

Applies and it is called odd if for all

Applies. Clearly a real function is exactly then just when you graph function is axially symmetric to the y- axis, and odd if their function graph is point symmetric to the origin.

Examples

Especially functions

  • The constant function
  • The absolute value function
  • The standard parabola
  • The cosine function
  • The Sekansfunktion
  • The bell curve

Odd functions

  • The sign function
  • The identical function
  • The cubic function
  • The sine function
  • The tangent function
  • The Gaussian error function

The only function that is even and odd at the same time, is the zero function.

More general examples

  • A power function
  • A polynomial function
  • A trigonometric polynomial

Decomposition

In which

The straight portion of the function and

Is the odd part of the function.

Properties

Algebraic properties

  • Each multiple of an even or odd function is even or odd again.
  • The sum of two even functions is straight again.
  • The sum of two odd functions is odd again.
  • The product of two even functions is straight again.
  • The product of two odd functions is even.
  • The product is an even and an odd function is an odd number.
  • 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

  • The zero point ( unless it is contained in the domain ) any odd function the function value is zero.
  • The derivative of a straight differentiable function is odd, the derivative of a differentiable function just odd.
  • The definite integral of an odd continuous function results when the integration limits are symmetrical around the zero point.
  • The Taylor series with the development point of even (odd ) function contains only even (odd ) powers.
  • The Fourier series of an even (odd ) function includes only the cosine ( sine ) terms.

Generalizations

More generally defined in algebra by the above definition, even and odd functions between two quantities and on which a link is given by additive inverse, such as (additive ) groups, rings, fields and vector spaces. In this way, for example, odd and even complex functions or even and odd vector-valued functions define.

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

Reflection symmetry Domain of a function Inverse element Harro Heuser Wolfram Research PlanetMath
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