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PAPER TITLE: COMPLEX ANALYSIS

EXAM DATE: FRIDAY 12, JUNE 2007

COURSE CODE: M337/K

Question 1

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

(a)  

(b) the principal cube root of  

(c)

(d)

ANSWERS(Purchase full paper to get all the solutions)

1a)

 

Rationalize

 

 

 

1b)

 

 

         =

 

 

1c)

 

 

 

 

Therefore,

 

 

1d)

 

 

 

 

 

 

Question 2

Let A = {z : 1 < |z − i| < 2} and B = {z : | Re z| ≤ 3, |Im z| ≤ 1}.

(a) Make separate sketches of the sets A, B and A − B.

(b) Write down which of the four sets A, B and A − B, if any, is

(i) a region

(ii) a simply connected region

(iii) closed

(iv) compact.

Question 3

In this question Γ is the line segment from i to 1-i

(a) (i) Determine the standard parametrization for the line segment Γ.

(ii) Evaluate

.

(b) 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) = z5 + 3z2+i.

(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

  + 3z²+i=0

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

Question 7

Let q(z) =  be a velocity function on C − {0}.

(a) Explain why q represents a model fluid flow on C − {0} .

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

(c) Evaluate the flux of q across the unit circle {z : |z| = 1}.

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 C-{0}.

(ii) Describe the effect of g on a small disc centred at i

(iii) and  be the paths meeting at i and −i given by

:  =   

 : ) =  

Sketch these paths on a single diagram, clearly indicating their directions

(iv) Using part (b)(ii), or otherwise, sketch the directions of g() and g() at g(i).

(v) Explain why g is not conformal at 0.

Question 10

Let f be the function

 

(a) Use the Taylor series about 0 for sin z and  to show that the Laurent series about 0 for f is

 

Hence evaluate the integral

           

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

(b) Write down the domain A of f. Use the Uniqueness Theorem to show that f is the only analytic function with domain A such that

             for y ∈ R, y > 0.

(c) Show that f has singularities at points of the form kπ, k ∈ Z, and classify these singularities. (Hint: You may find it helps to use sin z = (−1 sin(z − kπ), where k ∈ Z.)

QUESTION 11

(a) show that

                             

and find all points at which the maximum is attained,

(b) Show that the functions

                             (|z| < 5)

and

                            g(z) =  (|z| > 5)

are indirect analytic continuations of each other.

Question 12

(a) Determine the extended Möbius transformation  that maps

 i to 0, ∞ to 1  and -i to ∞.

(b) Let

                             

                           

                           

(i) Sketch the regions R, S and T.

(ii) Explain why  (R) = S.

((iii) Hence determine a one-one conformal mapping  from R onto T.

(iv) Determine a one-one conformal mapping g from R onto the open unit disc D = {z : |z| < 1}. (There is no need to simplify your answer.)

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