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 z_0, the centre is called an isolated singularity. The disk is called a punctured disc and is represented 0 < | z – z_0 | < r.

Examples

f(z) = 1/z has an isolated singularity at z_0 = 0

f(z) = 1/sin z has (multiple) isolated singularities at z_0 = 0 , +/- \pi,   +/- 2\pi etc.

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

Counter examples

f(z)  = \sqrt{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. a_k = 0 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 a_{-N} \neq 0 but a_k = 0 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

z_0 is a removable singularity if its Laurent series centred at z_0 satisfies a_k = 0 for all k < 0 (i.e. negative powers).

f(z) = \frac{sin z }{ z} = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - .. 0 < |z| < \infty

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 >>

z_0 is an isolated singularity of f. z_0 is a removable singularity if and only if f is bounded near z_0.

Slide 8 Poles

\frac{sin z}{ z^5} has order 4. z_0 is a pole if and only if function approaches infinity as z approaches z_0.

Note if f(z) has a pole at z_0 then 1/f(z) has a removable singularity at z_0.

Slide 9 Essential singularities

e.g. e^{(\frac{1}{z})} = 1 + \frac{1}{z} + \frac{1}{2!} \frac{1}{z^2} + \frac{1}{3!} \frac{1}{z^3} + ...

has an essential singularity at z_0 = 0.

Consider z is a real number.

e^{(\frac{1}{z})}  approaches infinity as z approaches 0 from the right (the positive side) because 1/z is getting large and positive.

Whereas

e^{(\frac{1}{z})} 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 z_0 (zero in this case)

<< casorati weierstrass theorem >>

Slide 10 casorati weierstrass example I

Slide 11 casorati weierstrass example II

Slide 12 Picards Theorem

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One Response to “Coursera Complex Analysis Lesson 7.2 Isolated Singularities”


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


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