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 $f_{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}$ 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 $\mathbb{R}$	the sum of the monomials of even degree in that polynomial
exponential function $e^x$	all of $\mathbb{R}$	hyperbolic cosine function $\cosh$