Real Analysis/Landau notation

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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[edit | edit source]

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 | edit source]

Let and let

Let

If then we say that

"As , "

Examples[edit | edit source]

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

The Big-O[edit | edit source]

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 | edit source]

Let and let

Let

If there exists such that then we say that

"As , "

Examples[edit | edit source]

  • As ,
  • As ,

Applications[edit | edit source]

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

Differentiability[edit | edit source]

Let and .

Then is differentiable at if and only if

There exists a such that as , .

Mean Value Theorem[edit | edit source]

Let be differentiable on . Then,

As ,

Taylor's Theorem[edit | edit source]

Let be n-times differentiable on . Then,

As ,