School of Mathematics and StatisticsMAST30021 Complex Analysis, Semester 2 2022Written assignment 3 and Cover SheetStudent Name Student NumberSubmit your assignment solutions together with this coversheet via the MAST30021 Gradescopewebsite before Tuesday 20 September (noon AET) sharp. No extensions will be granted! Onlyexemptions 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 andincorrect 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 indicatedlectures 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 thesequestions.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 thetutor 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 51. Mandatory Summary 10 points.Write a summary of three lectures chosen from Lecture 14 to Lecture 19. Note, that any Theorem andDefinition, especially those with a name of a renowned mathematician, are worth mentioning. Use thespace below. Clearly indicate which lecture you are writing about.Lecture :Please, turn the page for the other questions!Page 2 of 5Lecture :Please, turn the page for the other questions!Page 3 of 5Lecture :Please, turn the page for the other questions!Page 4 of 52. 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. Moreovercompute the residues at all isolated singularities (removable singularities are excluded). You are allowedto use the fact that all zeros of sin(z) in the complex plane lie at z = pin with n ∈ Z. Simplify yourresults 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? z2cos(pi/z)(make use of the limit formula when computing the residues).3. Mandatory Question (simple proof) 10 points.Consider the real functionf(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 ofLecture 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 withn ∈ Z.)(b) Prove that the union of the open discs of convergence D(x0, R(x0)) of these Taylor series is equalto the setS = {z ∈ C|Re(z) 6= 0} ∪ {iy ∈ C| ? 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 functionf(z) =ezz.What are the inner and outer radii of convergence Ri via holomorphicity of the function why these are the radii)? (Hint: it is sometimes helpful to make useof 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)| ≥ Kwhenever |z| ≥ R, then f(z) is a polynomial.(Hint: Combine the theorems and corollaries in Lectures 14 and 15. Mention those explicitly!)Page 5 of 5