Real Analysis/Landau notation
←Dedekind's Construction | Real Analysis Landau notation |
Bibliography→ |
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 edit
The little-o provides a function that is of lower order of magnitude than a given function, that is the function is of a lower order than the function . Formally,
Definition edit
Let and let
Let
If then we say that
"As , "
Examples edit
- As , (and )
- As , (and )
- As ,
The Big-O edit
The Big-O provides a function that is at most the same order as that of a given function, that is the function is at most the same order as the function . Formally,
Definition edit
Let and let
Let
If there exists such that then we say that
"As , "
Examples edit
- As ,
- As ,
Applications edit
We will now consider few examples which demonstrate the power of this notation.
Differentiability edit
Let and .
Then is differentiable at if and only if
There exists a such that as , .
Mean Value Theorem edit
Let be differentiable on . Then,
As ,
Taylor's Theorem edit
Let be n-times differentiable on . Then,
As ,