# Difference between revisions of "Even part"

From Calculus

(→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 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. The other part, , is the odd part of .

## 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 |