TITLE: Why it is important to write a function as sum of even and odd functions?
QUESTION [66 upvotes]: For the function $f(x)$ we can write it as sum of even and odd functions:
$$f(x)=\underbrace{\frac{f(x)+f(-x)}{2}}_{\text{Even}}+\underbrace{\frac{f(x)-f(-x)}{2}}_{\text{Odd}}$$
My question is why it is important for us to write a function as sum of these two even and odd functions? Is there any application of that?

REPLY [16 votes]: A famous example of a decomposition in odd and even functions is given by Euler's formula
\begin{align*}
\color{blue}{e^{iz}}&\color{blue}{=}\color{blue}{\cos z+i\sin z}\\
&=\frac{e^{iz}+e^{-iz}}{2}+\frac{e^{iz}-e^{-iz}}{2}\qquad\qquad z\in\mathbb{C}\\
\end{align*}
which is used in many applications.