An odd function
satisfies f(x) = -f(-x) and even function
s satisfy f(x) = f(-x).
These kinds of functions are very useful*
because of their inherent symmetry
Amazingly any old non-symmetric
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)
f(-x) = fe(-x) + fo(-x) = fe(x) - fo(x)
Solving continuously gives:
fe(x) = 1/2 (f(x) + f(-x))
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
is an odd function
, it's rather handy that we can represent any function
by summing a sine
part and a cosine