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 provides a function that is of lower order of magnitude than a given function, that is the function ${\displaystyle o(g(x))}$ is of a lower order than the function ${\displaystyle g(x)}$. Formally,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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