MAT3705 Assignment 4 (Accurate Solutions) 2025 Due 11 September 2025

MAT3705
Assignment 4
Due 11 September 2025

, MAT3705 Assignment 4 — Due date: 11 September 2025


MAT3705 Assignment 4
Due date: 11 September 2025


Question 1

Problem Statement. Let
1
f (z) = .
sin(1/z)

(a) What type of singularity is z = 0? Provide reasons.
1
(b) What type of singularity is z = 2π
? Provide reasons.

(a) Singularity at z = 0
Step 1. Substitute w = 1/z, so sin(1/z) = sin w. Expanding about w = 0:

w3 w5
sin w = w − + − ··· .
3! 5!

Thus
1 1 1
sin(1/z) = − 3+ − ··· .
z 6z 120z 5

Step 2. The Laurent expansion has infinitely many negative powers. Hence sin(1/z) has
an essential singularity at z = 0, and so does f (z) = 1/ sin(1/z).

Final Answer (a): z = 0 is an essential singularity.


1
(b) Singularity at z = 2π
1
Step 1. sin(1/z) = 0 when 1/z = kπ, i.e. z = 1/(kπ). Thus z = 2π
corresponds to
k = 2.

Step 2. sin w has simple zeros at w = kπ since cos(kπ) ̸= 0. Therefore sin(1/z) has
simple zeros at z = 1/(kπ).

Step 3. A reciprocal of a function with a simple zero gives a simple pole. So f (z) has a
1
simple pole at z = 2π
.

1
Final Answer (b): z = 2π
is a simple pole.


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