Difference between revisions of "Even part"

From Calculus
Jump to: navigation, search
(Definition)
(Definition)
 
Line 9: Line 9:
 
<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]].
+
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]]. The other part, <math>f_o</math>, is the [[odd part]] of <math>f</math>.
  
 
==Particular cases==
 
==Particular cases==

Latest revision as of 20:05, 22 September 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. 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