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]