# Even part

## Definition

Suppose $f$ is a function whose domain is a subset of the reals that is symmetric about 0, i.e., for every $x$ in the domain of $f$, $-x$ is also in the domain of $f$. Then, the even part of $f$, sometimes denoted $f_e$ or $f_{\operatorname{even}}$ is defined as a function with the same domain, and with the definition:

$\! f_e(x) := \frac{f(x) + f(-x)}{2}$

Equivalently, it is the only possible choice of even function in a decomposition of $f$ of the form:

$\! f(x) = f_e(x) + f_o(x)$

with $f_e, f_o$ both having the same domain as $f$, and with $f_e$ an even function and $f_o$ an odd function. The other part, $f_o$, is the odd part of $f$.

## Particular cases

Function Domain Even part
polynomial all of $\R$ the sum of the monomials of even degree in that polynomial
exponential function $e^x$ all of $\R$ hyperbolic cosine function $\cosh$