## Coursera Complex Analysis 6.3 Radius of convergence of a series

### Aug 11, 2016

## Slide 2 Determining Radius of Convergence for a power series. The Ratio Test

In earlier lectures we were told for a power series there is a value of R such that if distance of any z from the “centre” is less than R, i.e. | z – | < R then the power series converges whereas if |z – | > R the series diverges.

How do we find R.?

Theorem: **if** **reaches a limit** R as n tends to infinity, then this limit **is** the radius of convergence.

## Slide 3 Examples

, so = 0, a_k = 1, so then R = 1 (about the point z = 0).

, so = 0, a_k = k, , so R = 1

, so = 0, a_k = 1/k!, therefore R =

## Slide 4 Root Test << to be done >>

## Slide 5 Root Test examples << to be done >>

## Slide 6 Cauchy Hadamard

## Slide 7 Relationship Analytic Functions and Power series

If f: U -> C is analytic

and i.e. considering the z values in a disk of radius r surrounding the point << diagram >>, then in this disk f **HAS** a power series representation and that representation **is** and its radius of convergence is at least the radius of the disk surrounding , i.e. .

## Slide 8 Examples of Taylor series of exp(z) about different points in complex plane

Summary of Taylor series expansion

. Note carefully the expression for the coefficient.

consider , then all derivatives . I fin the Taylor expansion we set z_0 = 0, then for every derivative, thus for all k so

Similarly, about

## Slide 9 Series sin z about 0

Sin (z) is analytic in C. Then about 0

f(z) | sin(z) | f(0) | sin(0)=0 |

f’(z) | cos(z) | f’(0) | cos(0)=1 |

f’’(z) | -sin(z) | f’’(0) | -sin(0)=0 |

f’’’(z) | -cos(z) | f’’’(0) | -cos(0)= -1 |

f(4)(z) | sin(z) | f(4)(0) | sin(0)=0 |

so

=

## Slide 10 Series cos z about 0

Differentiating term by term

Cos z =

## Slide 11 Analytic function

An analytic function is determined by all its derivatives at the centre of the disc.