## Coursera Complex Analysis Lesson 7.2 Isolated Singularities

### Aug 16, 2016

## Slide 2

If a function is analytic **everywhere** on a disc **apart** from the point at its centre , the centre is called an **isolated singularity**. The disk is called a **punctured disc** and is represented 0 < | z – | < r.

### Examples

f(z) = 1/z has an isolated singularity at = 0

f(z) = 1/sin z has (multiple) isolated singularities at = 0 , +/- \pi, +/- 2\pi etc.

f(z) = 1/ (z-2) has isolated singularity at z = 2

Counter examples

f(z) = and Log(z) do not have isolated singularities at 0 since there is no punctured disc around 0 where the functions are analytic. The functions are not analytic on the negative real axis.

## Slide 3

As we have removed the isolated singularity from the disc, we have created an annulus around this point and hence the (analytic) function has a Laurent series.

## Slide 4 Behaviour of Laurent series near the isolated singularity

f(z) = (cos z – 1 )/ z^2 = -1/2! +z^2 /4! – …

**No** negative powers

f(z) = cos z / z^4 = 1/z^4 – (1/2!)(1/z^2) + (1/4!) –

**Finitely** many negative powers

f(z) = 1 / cos z = 1 – (1/2!)(1/z^2) + (1/4!)(1/z^4) – (1/6!)(1/z^6) + …

**Infinitely** many negative powers

## Slide 5 Definition according to number of negative powers of Laurent series

In the Laurent series surrounding an isolated singularity

If coefficients of negative powers are all zero (there are **no negative power** terms, i.e. for k < 0) in Laurent series then the singularity is **removable**.

If there are **finitely** many negative terms in Laurent series around singularity then these singularities also called **poles**. i.e there exists an N > 0, such that but for all k < -N. N is the order of the pole. if N = 1 this is a simple pole.

If there are **infinitely** many negative powers in Laurent series around singularity then this is an **essential** singularity. i.e. a_{k} \neq 0$ for infinitely many k < 0.

## Slide 6 Summary table of type of singularity based on Laurent expansion and number of negative power indices

## Slide 7 Removable singularities

is a removable singularity if its Laurent series centred at satisfies = 0 for all k < 0 (i.e. negative powers).

0 < |z| <

which looks like a Taylor series

and here if we **define** f(0) =1

f has **become analytic** in C and the singularity has been **removed.**

<< Statement of Riemann Theorem >>

is an **isolated** singularity of f. is a **removable** singularity if and only if **f is bounded** near .

## Slide 8 Poles

has order 4. is a pole if and only if function approaches infinity as z approaches .

Note if f(z) has a pole at then 1/f(z) has a removable singularity at .

## Slide 9 Essential singularities

e.g.

has an essential singularity at = 0.

Consider z is a real number.

approaches infinity as z approaches 0 from the right (the positive side) because 1/z is getting large and positive.

Whereas

approaches 0 as z approaches 0 from the left (the negative side) because 1/z is getting large and negative.

So f does not have a limit as z approaches the isolated singularity (zero in this case)

<< casorati weierstrass theorem >>

Aug 19, 2016 at 10:17 am

[…] is an essential singularity (see) because infinitely many terms with negative […]