, MAT3705 ASSIGNMENT 4 2025 SOLUTIONS
DUE DATE: 11 SEPTEMBER 2025
Question 1
Let
𝟏
(𝒇(𝒛) = )
𝐬𝐢𝐧(𝟏/𝒛)
(a) What type of singularity is ( 𝒛 = 𝟎 )? Provide reasons for your answer.
Solution:
• Zeros of sin(𝑤 ) occur when 𝑤 = 𝑛𝜋, 𝑛 ∈ 𝑍.
• For 𝑓 (𝑧), zeros of the denominator happen when
𝑧1 = 𝑛𝜋 ⟹ 𝑧 = 𝑛𝜋1
1 1 1
• Hence 𝑓 (𝑧) has poles at 𝑧 = , , , … , which accumulate at 𝑧 = 0.
π 2π 3π
• Therefore, 𝑧 = 0 is not an isolated singularity but an essential singularity.
𝒛 = 𝟎 is an essential singularity (non-isolated).
𝟏
(b) What type of singularity is 𝒛 = ? Provide reasons for your answer.
𝟐𝜋
Solution:
1
• At 𝑧 = :
2𝜋
sin(𝑧1 ) = sin(2𝜋) = 0.
• Near this point, expand:
sin(𝑧1 ) ≈ 𝑧1 − 2𝜋.
DUE DATE: 11 SEPTEMBER 2025
Question 1
Let
𝟏
(𝒇(𝒛) = )
𝐬𝐢𝐧(𝟏/𝒛)
(a) What type of singularity is ( 𝒛 = 𝟎 )? Provide reasons for your answer.
Solution:
• Zeros of sin(𝑤 ) occur when 𝑤 = 𝑛𝜋, 𝑛 ∈ 𝑍.
• For 𝑓 (𝑧), zeros of the denominator happen when
𝑧1 = 𝑛𝜋 ⟹ 𝑧 = 𝑛𝜋1
1 1 1
• Hence 𝑓 (𝑧) has poles at 𝑧 = , , , … , which accumulate at 𝑧 = 0.
π 2π 3π
• Therefore, 𝑧 = 0 is not an isolated singularity but an essential singularity.
𝒛 = 𝟎 is an essential singularity (non-isolated).
𝟏
(b) What type of singularity is 𝒛 = ? Provide reasons for your answer.
𝟐𝜋
Solution:
1
• At 𝑧 = :
2𝜋
sin(𝑧1 ) = sin(2𝜋) = 0.
• Near this point, expand:
sin(𝑧1 ) ≈ 𝑧1 − 2𝜋.
