B.6 EVEN AND ODD FUNCTIONS
Definition: Even and Odd Even function gE(x) and odd function gO(x) are defined by having the following properties about x = 0:
(B.29)
Any function g(x) can be decomposed as follows:
(B.30)
where
An example of this decomposition is shown in Figure B.6. The only type of function that is even and odd, satisfying both equations in (B.31), is a constant function.
Several properties of even and odd functions are provided in Table B.1 without proof. It is easy to see that for finite a:
Function | Property |
g1E(x)g2E(x) | Even |
g1O(x)g2O(x) | Even |
gE(t)gO(x) | Odd |
g1E(x)+g2E(x) | Even |
g1O(x)+g2O(x) | Odd |
gE(t)+gO(x) | Neither |
dgE(x)/dx | Odd |
dgO(x)/dx | Even |
(B.32)
which is useful when calculating moments of a distribution. When an integral of the following form is obtained
(B.33) ...
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