Even part

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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