MATH2036
COMPLEX FUNCTIONS
FINAL EXAM
A LEVEL 2 MODULE, 2021-2022
Problem 1.
(a) A set A ⊂ C is called convex if ∀x, y ∈ A and ∀λ ∈ [0, 1] [12 marks]
(1 − λ)x + λy ∈ A.
Show that if domains An, n = 1, 2, . . . are convex and ∩∞n=1An = ∅ then the set ∪∞n=1An is a starshape-domain.
(b) Let the function f be given by
f(x, y) = x(x
2 − 3y
2 − 2y) + i(3x
2
y − y
3 + x
2 − y
2
).
(i) Apply Cauchy-Riemann theorem to show that the function f is an entire function. [6 marks]
(ii) Use your computations from (i) to compute f
0 (z) and f(z) as functions of z. [7 marks]
Problem 2.
(a) Let γ = γ1 ∪ γ2 be the piecewise smooth contour, with γ1 being the straight line segment from 0 to i, and γ2 being the arc of the circle centered at 0 from i to −1 counter-clockwise. Evaluate the integral
and write your final answer in the form. A + Bi. [12 marks]
(b) Consider the series
(i) Determine the radius of convergence. [4 marks]
(ii) What is the value f
000 (0)? [4 marks]
(iii) Determine the following integral once counter-clockwise around the circle [5 marks]
Problem 3.
(a) Consider the function [10 marks]
and determine the following:
(i) the Laurent series of f in {3 < |z| < 5},
(ii) the Laurent series of f in {0 < | z − 3| < 2}.
(b) With f given as in (a) determine the following integrals (once counter-clockwise around the given circle). [9 marks]
(i)
(ii)
(iii)
(c) For f given as in (a) and [6 marks]
show that |I| ≤ 3/16π.
Problem 4.
(a) Let
(i) Determine the location of the singularities of f and the residues at each of the singularities. [6 marks]
(ii) For R > 0 let σR denote the straight line segment connecting −R and R, and let γR denote the semicircle connecting R with −R via the point iR. Let ΓR be the closed PSC that consists of σR followed by γR. Determine the values of R > 0 for which the integral [6 marks]
is defined and compute the value of IR whenever it is defined.
(iii) Show that [7 marks]
and determine the value of
(b) Show that if f is analytic in a domain D satisfying [6 marks]
then f must be a constant function.