Odd part - Revision history

Vipul: Created page with "==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 $f$
2011-09-22T21:18:09Z

Created page with "==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</mat..."

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==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 '''odd part''' of <math>f</math>, sometimes denoted <math>f_o</math> or <math>f_{\operatorname{odd}}</math> is defined as a function with the same [[domain]], and with the definition:

<math>\! f_o(x) := \frac{f(x) - f(-x)}{2}</math>

Equivalently, it is the only possible choice of [[defining ingredient::odd function]] in a decomposition of <math>f</math> of the form:

<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]]. The other part, <math>f_e</math>, is the [[even part]] of <math>f</math>.

==Facts==

===Effect on even and odd functions===

* The odd part of an even function is the [[zero function]] on the same domain.
* The odd part of an odd function is the same function.

===Properties preserved on taking the odd part===

{| class="sortable" border="1"
! Function property !! Explanation for why taking the odd part preserves this property
|-
| [[continuous function]] || [[odd part of continuous function is continuous]]
|-
| [[differentiable function]] || [[odd part of differentiable function is differentiable]]
|-
| [[periodic function]] || [[odd part of periodic function is periodic]]
|-
| [[infinitely differentiable function]] || [[odd part of infinitely differentiable function is infinitely differentiable]]
|-
| [[analytic function]] || [[odd part of analytic function is analytic]]
|}

===Effect of operators on odd part===

{| class="sortable" border="1"
! Operator !! Effect on odd part (short version) !! Effect on odd part (in symbols) !! Proof
|-
| [[pointwise sum of functions|pointwise sum]] || the odd part of a sum of functions is the sum of the odd parts of each function || <math>(f + g)_o = f_o + g[[odd part operator is linear]]
|}
==Particular cases==

{| class="sortable" border="1"
! Function !! Domain !! Odd part
|-
| [[polynomial function]] || all reals || sum of all the monomials of odd degree in the polynomial
|-
| [[exponential function]] <math>e^x</math> || all reals || [[hyperbolic sine function]] <math>\sinh</math>
|}