MTH 219 Complex Functions Coursework 2 AY 2024/2025
Problems
1. Find the residue of the function z3ez/1 at z = 0 and show that it has no anti-derivative on the punctured complex plane C \ {0}. (18 marks)
2. Find the Laurent series of z2 - 3z + 2/1 valid in following the regions respectively: (20 = 10 + 10 marks)
A1 = {z ∈ C | 0 < |z − 1| < 1} and A2 = {z ∈ C | 1 < |z| < 2}.
3. Locate the singularities off f(z) = ez + 1/(z2 + π2)2 and determine their types. If a singularity is a pole, find its order. (16 marks)
4. Solve the following problem. (16 = 6 + 10 marks)
(a) State Liouville’s theorem.
Show that the function g(z) = z/sinz, is not bounded on C \ {0}.
5. Let f (z) = z4 + 2/z2. (20 = 5 + 5 + 5 + 5 marks)
(a) Let R > 0 is a real number. Sketch the contour C = L + γR , where γR (t) = Reit , 0 ≤ t ≤ π , and L(t) = t, −R ≤ t ≤ R.
(b) Show that γR f (z)dz → 0 if R → ∞ .
(c) Assume that R > 2. Calculate C f (z)dz. (Hint: Use Cauchy’s residue theorem.)
Using the results obtained in (a)-(c), evaluate the improper integral x4 + 2/x2dx.
Total mark: 100 marks = Problems (90 marks) + Clarity of computations and mathematical reasoning, as well as neatness of overall presentation (10 marks).