Difference between revisions of "Even part"

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(Definition)
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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:
 
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:
 
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]].
 
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]].

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