School of Mathematics and Statistics
MAST30021 Complex Analysis, Semester 2 2022
Written assignment 3 and Cover Sheet
Student Name Student Number
Submit your assignment solutions together with this coversheet via the MAST30021 Gradescope
website before Tuesday 20 September (noon AET) sharp. No extensions will be granted! Only
exemptions for suitably justified reasons can be granted.
This assignment is worth 5% of your final MAST30021 mark.
Assignments must be neatly handwritten unless you have a medical exception.
Full working must be shown in your solutions.
Marks will be deducted in every question for incomplete working, insufficient justification of steps and
incorrect mathematical notation.
You must use methods taught in MAST30021 Complex Analysis to solve the assignment questions.
The first task is mandatory for everyone, meaning everyone must write a summary of three indicated
lectures of not less than half and not more than one entire A4 page for each lecture.
The second and third tasks are mandatory for everyone, meaning everyone must completely answer these
questions.
The fourth and fifth tasks are optional, meaning you have to decide which of the two you want to answer.
In the case you answer both, indicate which of the two questions should be marked! If you do not do that the
tutor has free choice to decide which is going to be marked.
There are in total 40 points to achieve. You cannot get 50 points!
Begin your answer for each question on a new page!
Please, turn the page for the other questions!
Page 1 of 5
1. Mandatory Summary 10 points.
Write a summary of three lectures chosen from Lecture 14 to Lecture 19. Note, that any Theorem and
Definition, especially those with a name of a renowned mathematician, are worth mentioning. Use the
space below. Clearly indicate which lecture you are writing about.
Lecture :
Please, turn the page for the other questions!
Page 2 of 5
Lecture :
Please, turn the page for the other questions!
Page 3 of 5
Lecture :
Please, turn the page for the other questions!
Page 4 of 5
2. Mandatory Question (simple computation) 10 points.
Find all zeros and singularities of the following functions and classify those (isolated or not, essential,
removable or the order of the poles and zeros). Give an explanation of your classification. Moreover
compute the residues at all isolated singularities (removable singularities are excluded). You are allowed
to use the fact that all zeros of sin(z) in the complex plane lie at z = pin with n ∈ Z. Simplify your
results as much as possible (fractions and factors of pi remain as they are)!)
(a) f(z) =
sin2(z)
sinh2(z)
(make use of Landau symbols when computing the residue),
(b) f(z) =
4? z2
cos(pi/z)
(make use of the limit formula when computing the residues).
3. Mandatory Question (simple proof) 10 points.
Consider the real function
f(x) = ln(x2 + 1), with x ∈ R.
(a) Compute the Taylor series at any x0 ∈ R and show that its radius of convergence is R(x0) =√
1 + x20. When computing the radius of convergence make use of the theorems and corollaries of
Lecture 13 (say which you use and why you can apply those!) so that no ?N criterion is required.
(Hint: you can use the fact that the sequence {| cos(j?)|}j∈N is divergent for any ? 6= npi/2 with
n ∈ Z.)
(b) Prove that the union of the open discs of convergence D(x0, R(x0)) of these Taylor series is equal
to the set
S = {z ∈ C|Re(z) 6= 0} ∪ {iy ∈ C| ? 1 < y < 1}.
4. Optional Question (advanced computation) 10 points.
Compute the Laurent series at any point z0 ∈ C and any open annulus centred at z0 of the function
f(z) =
ez
z
.
What are the inner and outer radii of convergence Ri < Ro (you only need to give a simple explanation
via holomorphicity of the function why these are the radii)? (Hint: it is sometimes helpful to make use
of the contour integral formula to find the Laurent coefficients!)
5. Optional Question (advanced proof) 10 points.
Prove the following statement from Lecture 15: If f(z) is entire and there are R,K > 0 with |f(z)| ≥ K
whenever |z| ≥ R, then f(z) is a polynomial.
(Hint: Combine the theorems and corollaries in Lectures 14 and 15. Mention those explicitly!)