## Coursera Complex Analysis 6.2 Taylor Series

### Aug 9, 2016

## Slide 2 Definition Taylor series

Series of form, Centred at z_0 \in C.

Examples

, converges for |z| < 1

where w =

this converges when

converges when and diverges when

## Slide 3 Radius Convergence Theorem *important *

Let be a power series.

There exists a real number R, , such that the series

converges absolutely in **<** R

and diverges in **>** R

convergence is uniform for each r < R

## Slide 4 Examples Radius Convergence

Pick an arbitrary

Note:

But eventually this will happen, no matter what the value of |z| is, k will cause the terms of the series to be increasing in size and the series does not converge.

In contrast: . Similarly to above pick an arbitrary z, then when k reaches the value and beyond this value of k all subsequent terms are less than powers of 1/2 so the series will converge. Since this is always true the series has infinite radius of convergence.

## Slide 5 Analycity of Power Series

Theorem. For the power series with radius of convergence R > 0, the claim is:

f(z) = is analytic in

and since it is analytic it can be differentiated (infinitely) as follows (note change of limits):

f ‘ (z) =

f ” (z) =

eventually

(since it makes sense to define ).

Then by rearranging we get an expression for every coefficient in the original power series

## Slide 6 Differentiating term-by-term

Because with radius of convergence 1, the sum is analytic when |z| < 1

So we can differentiate term by term and end up with

## Slide 7 Integrating term-by-term

Similarly , with certain certain conditions (to be added), can integrate infinite series term by term.

## Slide 8 Integration example to get series for Log(z)

Justification that when |z|< 1.

By fiddling with variable we can get: for |z – 1| < 1, we have a power series for Log.