An
odd function satisfies f(x) = -f(-x) and
even functions satisfy f(x) = f(-x).
These kinds of functions are very useful
* because of their
inherent symmetry.
Amazingly any old
non-symmetric,
weird,
common or
garden function can be represented as the
sum of an
odd function and an
even function.
Functions which always work in the above:
Let f(x) be an arbitrary function.
Now, assume that there exists an even function fe(x) and an odd function fo(x) such that
f(x) = fe(x) + fo(x)
even odd
then
f(-x) = fe(-x) + fo(-x) = fe(x) - fo(x)
Solving continuously gives:
fe(x) = 1/2 (f(x) + f(-x))
and
fo(t) = 1/2 (f(x) – f(-x)) !
*Why is this useful?
Well...
Fourier analysis seeks to represent
arbitrary functions by infinite series of
sinusoids. Since
sine is an
even function and
cosine is an
odd function, it's rather handy that we can represent
any function by summing a
sine part and a
cosine part!