Real Analysis/Landau notation

The Landau notation is an amazing tool applicable in all of real analysis. The reason it is so convenient and widely used is because it underlines a key principle of real analysis, namely estimation. Loosely speaking, the Landau notation introduces two operators which can be called the "order of magnitude" operators, which essentially compare the magnitude of two given functions.

The little-o

The little-o provides a function that is of lower order of magnitude than a given function, that is the function $o(g(x))$ is of a lower order than the function $g(x)$ . Formally,

Definition

Let $A\subseteq \mathbb {R}$ and let $c\in \mathbb {R}$

Let $f,g:A\to \mathbb {R}$

If $\lim _{x\to c}{\frac {f(x)}{g(x)}}=0$ then we say that

"As $x\to c$ , $f(x)=o(g(x))$ "

Examples

As $x\to \infty$ , (and $m<n$ ) $x^{m}=o(x^{n})$
As $x\to \infty$ , (and $n\in \mathbb {N}$ ) $\ln x=o(x^{n})$
As $x\to 0$ , $\sin x=o(1)$

The Big-O

The Big-O provides a function that is at most the same order as that of a given function, that is the function $O(g(x))$ is at most the same order as the function $g(x)$ . Formally,

Definition

Let $A\subseteq \mathbb {R}$ and let $c\in \mathbb {R}$

Let $f,g:A\to \mathbb {R}$

If there exists $M>0$ such that $\lim _{x\to c}\left|{\frac {f(x)}{g(x)}}\right|<M$ then we say that

"As $x\to c$ , $f(x)=O(g(x))$ "

Examples

As $x\to 0$ , $\sin x=O(x)$
As $x\to {\tfrac {\pi }{2}}$ , $\sin x=O(1)$

Applications

We will now consider few examples which demonstrate the power of this notation.

Differentiability

Let $f:{\mathcal {U}}\subseteq \mathbb {R} \to \mathbb {R}$ and $x_{0}\in {\mathcal {U}}$ .

Then $f$ is differentiable at $x_{0}$ if and only if

There exists a $\lambda \in \mathbb {R}$ such that as $x\to x_{0}$ , $f(x)=f(x_{0})+\lambda (x-x_{0})+o\left(|x-x_{0}|\right)$ .

Mean Value Theorem

Let $f:[a,x]\to \mathbb {R}$ be differentiable on $[a,b]$ . Then,

As $x\to a$ , $f(x)=f(a)+O(x-a)$

Taylor's Theorem

Let $f:[a,x]\to \mathbb {R}$ be n-times differentiable on $[a,b]$ . Then,

As $x\to a$ , $f(x)=f(a)+{\tfrac {(x-a)f'(a)}{1!}}+{\tfrac {(x-a)^{2}f''(a)}{2!}}+\ldots +{\tfrac {(x-a)^{n-1}f^{(n-1)}(a)}{(n-1)!}}+O\left((x-a)^{n}\right)$