1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

7.4 • SINGULARITIES, ZEROS, AND POLES 277

A (a, 0, R). We now look at this special case of Laurent's theorem in order to
classify three types of isolated singularities.

Definition 7 .5: Classification of singularities

Let f have an isolated singularity at a with Laurent series

00

f (z) = L Cn(z- a)n (valid for all z EA (a, 0, R) ).

n=- oo

Then we distinguish the following types of singularities at a.

i. If Cn = 0, for n = -1, -2, -3, ... , then f has a removable singularity at

a.

ii. If k is a positive integer such that C-1< =f 0, but c,., = 0 for n < -k, then f

has a pole of order k at a.

iii. If c,., =f 0 for infinitely many negative integers n, then f has an essential

singularity at a.

Let's investigate some examples of these three cases.

i. If f has a removable singularity at a, then it has a Laurent series

00

f(z)= :Lcn(z-at (valid for all z EA (a, 0, R) ).

n=O

Theorem 4.17 implies that the power series for f defines an analytic function

in the disk DR (a). If we use this series to define f (a)= eo, then the function

f becomes analytic at z = a, removing the singularity. For example, consider

the function f (z) = si~z. It is undefined at z = 0 and has an isolated

singularity at z = 0 because the Laurent series for f is

We can remove this singularity if we define f (0) = 1, for then f will be

analytic at 0 in accordance with Theorem 4.17.