Difference between revisions of "Even part"
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<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 
|
