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even and odd functions

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

Mathematics Subject Classification

26A06 no label found

Comments

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.

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).

Please file corrections rather than posting messages.

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