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PAPER TITLE: COMPLEX ANALYSIS

EXAM DATE: THURSDAY 15, JUNE 2009

COURSE CODE: M337/H

Question 1

Determine each of the following complex numbers in Cartesian form, simplifying your answers as far as possible.

(a)  

(b)

(c)

(d) 

ANSWERS(Purchase full paper to get all the solutions)

1a)

 

 

 

1b)

 

 

 

 

 =

1c)

 

 

 

1d)

i  

Recall

 

 

 

 

Question 2

Let

A = {z : -1 } and B = {z : }.

(a) Make separate sketches of the sets A, B, and C = A - B and .

(b) For each of the sets A, B, and C.

(i) state whether it is a region, and if not explain why not

(ii) state whether it is compact, and if not explain why not.

Question 3

Let Γ be the line segment from -1 to i

(a) Write down the standard parametrization for Γ.

(b) Evaluate

.

(c) Determine an upper estimate for the modulus of

 

Question 4

Evaluate the following integrals in which C  = {z : |z| = 12 }

Name any standard results that you use and check that their hypotheses are satisfied.

1. 
2. 
3. 

Question 5

(a) Find the residues of the function

 

at each of the poles of f.

(b) Hence evaluate the integrals

                             

Question 6

Let f(z) =  + 4z³-2i .

(a) Determine the number of zeros of f that lie inside:

(i) the circle C₁ = {z : |z| = 2},

(ii) the circle C₂ = {z : |z| = 1}.

(b) Show that the equation

   + 4z³-2i  = 0

has exactly four solutions in the set {z : 1 < |z| < 2}

Question 7

Let q(z) =    be a velocity function.

(a) Explain why q represents a model fluid flow on .

(b) Determine a complex potential function for this flow. Hence sketch the streamline through the point 1 and indicate the direction of flow.

(c) Evaluate the circulation of q along the path

Γ : γ(t) = t (t ∈ [0, 2]).

Question 8

(a) Find the fixed points of the function  and classify them as (super-)attracting, repelling or indifferent.

(b) Which of the following points c lie in the Mandelbrot set.

(i) c =  

(ii) c =

Justify your answer in each case

Question 9

(a) Let f be the function

 

i) Write  where u and v are real-valued functions.

(ii) Use the Cauchy–Riemann theorem and its converse to show that f is differentiable, at    but not analytic there.

 

(b) Let g be the function   .

(i) Show that g is conformal at i.

(ii) Let  and  be the paths

:  =    

 : ) =

Show that  and  intersect at the points i and Sketch the paths  and  on the same diagram,

(iii) Describe the effect of g on a small disc centred at i and hence make a sketch showing the approximate directions of the paths of g( ) and g( ) near the point g(i).

Question 10

(a) Let f be the function

 

Write down the singularities of f and determine their nature.

(b) (i) Write down the Laurent series about 0 for the function

g(z) = sin(1/z),

giving an expression for the general term of the series and state its annulus of convergence.

     (ii) Hence evaluate the integral

,

where C is the unit circle {z : |z| = 1

(c) (i) Determine the first three non-zero terms of the Taylor series about 0 for the function

h(z) = Log(cos z).

(ii) Hence determine the first three non-zero terms of the Taylor series about 0 for tan.

Question 11

Let f be the function

 .

(a) (i) Find the residues of f at each of the points 0, i  and .

(b) Hence determine the sum of the series

 

(c) Use your solution to part (a)(ii) to prove that

           

Question 12

(a) Determine the extended Möbius transformation that maps

 1 to 0, -i to 1  and -1to ∞

(b) Let

                   R = {z : |z| < 1,Im z < 0}

and

                      S = {w : Im w < 0}.

(i) Sketch the regions R and S

(ii) Determine and sketch the image of R under  of part (a).

(iii) Hence determine a conformal mapping f from R onto S.

(iv) Write down the rule of the inverse function .

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