MATH2036-E1
A LEVEL 2 MODULE, SPRING SEMESTER 2022-2023
COMPLEX FUNCTIONS
1. (a) Consider the set
A = {z = x + iy : |y| < |x| + 1}
i) Sketch A and show that A is open. [6 marks]
ii) Show that A is a star-domain. [6 marks]
(b) Where is the function
f(x + iy) = (x3y — xy3) + ix2y2
complex differentiable? Where is the function analytic? [6 marks]
(c) Is there a domain D ⊂ C and a function F analytic in D such that RF(x + iy) = x2y for x + iy ∈ D? [7 marks]
2. (a) Let y = y1 ∪ y2 be a piecewise smooth contour, where y1
is the straight line segment from 0 to 1 + i, and y2
is the arc of the circle centered at 0 from 1 + i to 1 − i 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'''(0)? [4 marks]
iii) Determine the following integral once counter-clockwise around the circle
[5 marks]
3. (a) Consider the function
and determine the following:
i) the Laurent series of f in {1 < |z| < 2}, [5 marks]
ii) the Laurent series of f in {0 < |z − 1| < √5}. [5 marks]
(b) With f given as in (a) determine the following integrals (once counter-clockwise around the given circle)
i)
[3 marks]
ii)
[3 marks]
iii)
[3 marks]
(c) For f given as in (a) and
show that [6 marks]
4. (a) Let
i) Determine the location of the singularities of f and the residues at each of the singularities. [3 marks]
ii) For R > 0 let σR denote the straight line segment connecting −R and R, and let yR denote the semicircle connecting R with −R via the point iR. Let R be the closed PSC oriented counter-clockwise that consists of σR followed by yR. Determine the values of R > 0 for which the integral
is defined and compute the value of IR whenever it is defined. [6 marks]
iii) Show that
and determine the value of
[10 marks]
(b) Show that if f is analytic in a domain D satisfying
then f must be a constant function. [6 marks]