Real Analysis/Landau notation

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-oEdit

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,

DefinitionEdit

Let   and let  

Let  

If   then we say that

"As  ,  "

ExamplesEdit

  • As  , (and  )  
  • As  , (and  )  
  • As  ,  

The Big-OEdit

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,

DefinitionEdit

Let   and let  

Let  

If there exists   such that   then we say that

"As  ,  "

ExamplesEdit

  • As  ,  
  • As  ,  

ApplicationsEdit

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

DifferentiabilityEdit

Let   and  .

Then   is differentiable at   if and only if

There exists a   such that as  ,  .

Mean Value TheoremEdit

Let   be differentiable on  . Then,

As  ,  

Taylor's TheoremEdit

Let   be n-times differentiable on  . Then,

As  ,