## Coursera Complex Analysis Lesson 7.4 Finding Residues

### Aug 19, 2016

## IMPORTANT

## Slide 2 Restate Residue Theorem

- f has isolated singularity at and is analytic on the punctured disc 0 < | z- | < r
- f has a unique Laurent series representation …
- The residue of f at is represented by Res(f,) = the coefficient of

Following slides are to calculate residues at removable singularities and poles (not essential singularities?)

## Slide 3 Residue at Removable singularity

is a removable singularity of a Laurent series if there are no negative powers of the terms

so = 0

so Res(f,) = 0

## Slide 4 Residue at simple pole

(*)

How to isolate ? Cross-multiply then take limit as z tends to

We need to take limit because of the division by in the original Laurent series (*)

## Slide 5 Residue at simple pole example

(From previous slide)

Consider has simple poles at = +i, -i

## Slide 6 Residue at double pole example

Laurent series: (*)

is double pole when for k > 2.

To isolate . (i) Multiply by (z – z_0)^2 and (ii) differentiate (iii) take limit.

## Slide 7 Residue at double pole example

double pole at z = 1, simple pole at z = 3.

## Slide 8 Residue at poles of order n

Use method above: (i) Multiply by (z – z_0)^n and (ii) differentiate multiple times (iii) take limit.

Giving

Res(f,z_0) =

## Slide 9 Residue of “quotient” functions

f(z) =

Res(f,) =