Mathematics of Physics and Engineering

220 Singularities of Complex Functions

EXERCISE 4.4.2.C (a) Let ZQ be an isolated singular point of the function

f = f(z), and consider the corresponding Laurent series Y^'kL-oo ck(z—zo)k

converging for 0 < \z — zo\ < R. Verify that ZQ is

(i) a removable singularity if and only if Ck = 0 for allk < 0 Hint: this pretty

much follows from Theorem 4-3.4- Equivalently, ZQ is a removable singularity

if and only if there exists a 6 > 0 so that the function f(z) is analytic and

bounded for 0 < \z — ZQ\ < S.

(ii) a pole of order N if and only C-N ^ 0 and Ck = 0 for all k < —(N + 1)

Hint: for the proof in one direction, multiply the Laurent series by (z — zo)N; for

the proof in another direction, multiply f by (z — zo)N and use Theorem 4-3-4-

(Hi) an essential singularity if and only if Ck ^ 0 for infinitely many k < 0.

Hint: by definition, the essential singularity is the only remaining option.

(b) Verify that the point ZQ = 0 is

(i) a removable singularity for the functions f(z) = sin z/z,

f(z) = (ez — 1)^2 /(1 — cosz) and f(z) — (zcosz — sin z)/(z sin z);

(ii) a second-order pole for the function f(z) = (1 — cosz)/(ez - l)^4 ;

(Hi) an essential singularity for the function f(z) — e1//z;

(iv) not an isolated singularity for the function f(z) — l/sin(l/z).

As with the Taylor series, we usually do not use the formula (4.4.3) to

find the coefficients of the Laurent series, and use other methods instead.

FOR EXAMPLE, consider the function

««) —(I^I)-

This function is analytic everywhere except at the point ZQ = 1. Let us find

the Laurent series for / at ZQ. We have z/(z — 1) = (z - 1 + l)/(z — 1) =

1 + (z — 1)_1 and so

f(z) = sin(l + (z-l)-^1 ) =sinl cos((2 - 1)_1) + sin((z - l)-1) cosl,

where we use the formula for the sine of the sum. Then we use the standard

Taylor expansions for the sine and cosine to conclude that

f{Z) = Sln

' go <* - W*)'

+ C

°

S

%tz~ D^(2fc + 1)!

;

one could write this as a single series, but it will not add anything essential
to the final answer. We therefore conclude that ZQ — 1 is an essential
singularity of /.