# Difference between revisions of "Even part"

From Calculus

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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 is a function whose domain is a subset of the reals that is symmetric about 0, i.e., for every in the domain of , is also in the domain of . Then, the **even part** of , sometimes denoted or is defined as a function with the same domain, and with the definition:

Equivalently, it is the only possible choice of even function in a decomposition of of the form:

with both having the same domain as , and with an even function and an odd function.

## Particular cases

Function | Domain | Even part |
---|---|---|

polynomial | all of | the sum of the monomials of even degree in that polynomial |

exponential function | all of | hyperbolic cosine function |