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