Editing Even and odd functions (section)

==Even–odd decomposition==
If a real function has a domain that is self-symmetric with respect to the origin, it may be uniquely decomposed as the sum of an even and an odd function, which are called respectively the '''even part''' (or the '''even component''') and the '''odd part''' (or the '''odd component''') of the function, and are defined by
<math display="block">f_\text{even}(x) = \frac {f(x)+f(-x)}{2},</math>
and
<math display=block>f_\text{odd}(x) = \frac {f(x)-f(-x)}{2}.</math>

It is straightforward to verify that <math>f_\text{even}</math> is even, <math>f_\text{odd}</math> is odd, and <math>f=f_\text{even}+f_\text{odd}.</math>

This decomposition is unique since, if
:<math>f(x)=g(x)+h(x),</math>
where {{mvar|g}} is even and {{mvar|h}} is odd, then <math>g=f_\text{even}</math> and <math>h=f_\text{odd},</math>  since
: <math>\begin{align}
2f_\text{e}(x) &=f(x)+f(-x)= g(x) + g(-x) +h(x) +h(-x) = 2g(x),\\
2f_\text{o}(x) &=f(x)-f(-x)= g(x) - g(-x) +h(x) -h(-x) = 2h(x).
\end{align}</math>

For example, the [[hyperbolic cosine]] and the [[hyperbolic sine]] may be regarded as the even and odd parts of the exponential function, as the first one is an even function, the second one is odd, and 
:<math>e^x=\underbrace{\cosh (x)}_{f_\text{even}(x)} + \underbrace{\sinh (x)}_{f_\text{odd}(x)}</math>.
[[Joseph Fourier|Fourier]]'s [[sine and cosine transforms]] also perform even–odd decomposition by representing a function's odd part with [[sine waves]] (an odd function) and the function's even part with cosine waves (an even function).