# properties of $O$ and $o$

1. 1.

$o(f)$ and $O(f)$ are vector spaces, i.e. if $g,h\in o(f)$ (resp. in $O(f)$) then $\lambda g+\mu h\in o(f)$ (resp. in $O(f)$) whenever $\lambda,\mu\in\mathbb{R}$; In particular $o(f)+o(f)=o(f)$ and $\lambda o(f)=o(f)$;

2. 2.

if $\lambda\neq 0$ then $\lambda o(f)=o(f)$ and $\lambda O(f)=O(f)$;

3. 3.

$fo(g)=o(fg)$, $fO(g)=O(fg)$;

4. 4.

$o(g)^{\alpha}=o(g^{\alpha})$, $O(g)^{\alpha}=O(g^{\alpha})$;

5. 5.

$o(f)\subseteq O(f)$; on the other hand if $f\in o(g)$ then $O(f)\subseteq o(g)$;

6. 6.

$o(f)\subseteq o(g)$ if $f\in O(g)$; analogously $O(f)\subseteq O(g)$ if $f\in O(g)$;

7. 7.

$o(o(f))=o(f)$, $O(O(f))=O(f)$, $o(O(f))=o(f)$, $O(o(f))=o(f)$.

Here are some examples. First of all we consider Taylor formula. If $x_{0}\in(a,b)\subset\mathbb{R}$ and $f\colon(a,b)\to\mathbb{R}$ has $n$ derivatives, then

 $f(x)\in\sum_{k=0}^{n}\frac{f^{(k)}(x_{0})}{k!}(x-x_{0})^{k}+o((x-x_{0})^{n}).$

As a consequence, if $f$ has $n+1$ derivatives, we can replace $o((x-x_{0})^{n})$ with $O((x-x_{0})^{n+1})$ in the previous formula.

For example:

 $e^{x}\in 1+x+\frac{1}{2}x^{2}+\frac{1}{6}x^{4}+O(x^{5})\subset 1+x+\frac{1}{2}% x^{2}+\frac{1}{6}x^{4}+o(x^{4}).$

Using the properties stated above we can compose and iterate Taylor expansions  . For example from the expansions

 $\sin x\in x+\frac{x^{3}}{3!}+o(x^{4}),\qquad e^{x}\in 1+x+\frac{x^{2}}{2}+O(x^% {3}),$
 $\cos x\in 1-\frac{x^{2}}{2}+\frac{x^{4}}{4!}+o(x^{5})\subseteq 1-\frac{x^{2}}{% 2}+O(x^{4}),\qquad\log(1+x)\in x-\frac{x^{2}}{2}+o(x^{2})$

we get

 $\displaystyle(x\sin x-e^{(x^{2})})\log(\cos x)$ $\displaystyle\in\left(x(x-\frac{x^{3}}{3!}+o(x^{4}))-(1+x^{2}+\frac{x^{4}}{2}+% O((x^{2})^{3})\right)\log(1-\frac{x^{2}}{2}+\frac{x^{4}}{4!}+o(x^{5}))$ $\displaystyle=\left(x^{2}-\frac{x^{4}}{3!}+o(x^{4})-1-x^{2}-\frac{x^{4}}{2}+O(% x^{6})\right)\left(-\frac{x^{2}}{2}+\frac{x^{4}}{4!}+o(x^{5})-\frac{(-\frac{x^% {2}}{2}+o(x^{3}))^{2}}{2}+o((-\frac{x^{2}}{2}+o(x^{3}))^{2})\right)$ $\displaystyle=(-1-\frac{2}{3}x^{4}+o(x^{4})+O(x^{6}))\left(-\frac{x^{2}}{2}+% \frac{x^{4}}{4!}+o(x^{5})-\frac{\frac{x^{4}}{4}-2\frac{x^{2}}{2}o(x^{3})+(o(x^% {3}))^{2}}{2}+o(\frac{x^{4}}{4}+o(x^{4}))\right)$ $\displaystyle=(-1-\frac{2}{3}x^{4}+o(x^{4}))(-\frac{x^{2}}{2}+\frac{x^{4}}{4!}% +o(x^{5})+\frac{x^{4}}{8}+o(x^{5})+o(x^{6})+o(x^{4}))$ $\displaystyle=(-1-\frac{2}{3}x^{4}+o(x^{4}))(-\frac{x^{2}}{2}+6x^{4}+o(x^{4}))$ $\displaystyle=-\frac{x^{2}}{2}-6x^{4}+o(x^{4})+x^{4}O(x^{2})+o(x^{4})O(x^{2})$ $\displaystyle=-\frac{x^{2}}{2}-6x^{4}+o(x^{4})+O(x^{6})+o(x^{6})$ $\displaystyle=-\frac{x^{2}}{2}-6x^{4}+o(x^{4})$
Title properties of $O$ and $o$ PropertiesOfOAndO 2013-03-22 15:15:45 2013-03-22 15:15:45 paolini (1187) paolini (1187) 7 paolini (1187) Result msc 26A12 FormalDefinitionOfLandauNotation