Even part: Difference between revisions

Revision as of 12:43, 28 August 2011

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.

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 $\cosh$

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 Suppose <math>f</math> is a function whose domain is a subset of the reals that is symmetric about 0, i.e., for every <math>x</math> in the domain of <math>f</math>, <math>-x</math> is also in the domain of <math>f</math>. Then, the '''even part''' of <math>f</math>, sometimes denoted <math>f_e</math> or <math>f_{\operatorname{even}}</math> is defined as a function with the same [[domain]], and with the definition:
-<math>f_e(x) := \frac{f(x) + f(-x)}{2}</math>
+<math>\! f_e(x) := \frac{f(x) + f(-x)}{2}</math>
 Equivalently, it is the only possible choice of [[defining ingredient::even function]] in a decomposition of <math>f</math> of the form:
-<math>f(x) = f_e(x) + f_o(x)</math>
+<math>\! f(x) = f_e(x) + f_o(x)</math>
 with <math>f_e, f_o</math> both having the same domain as <math>f</math>, and with <math>f_e</math> an [[even function]] and <math>f_o</math> an [[odd function]].