# even and odd functions

## Primary tabs

Defines:
even function, odd function
Type of Math Object:
Definition
Major Section:
Reference

## Mathematics Subject Classification

### Properties

You could add some other properties:

The sum of even functions is even and the sum of odd functions is odd.
The only function that is even and odd is f(x)=0.
The Fourier transform of an real even function is purely real and even.
The Fourier transform of an real odd function is purely imaginary and odd.

### more general definition

The definition should be generalized to functions not defined on the whole of R, but on a symmetric set; think e.g. of the tan() function,
or of inverse trigonometric functions such as arcsin().

Two other possibilities for generalizations would be:

a) even / odd symmetry w.r.t. a point other than the origin
(where even symmetry is given when the graph is symmetric about
a parallel to the y-axis, and odd symmetry refers to a graph
symmetric about a point (x,y=f(x)).

b) functions defined on, and taking values in, arbitrary groups
(cf. SymmetricSet.html).

### Re: more general definition

Please file corrections rather than posting messages.