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PAPER TITLE: COMPLEX ANALYSIS

EXAM DATE: TUESDAY 18, OCTOBER 2005

COURSE CODE: M337/Q

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)

  

                 =

 

1b)

principal cube root of  

 

Recall

 

 

 

         =

         =

         =

Therefore,

The principal cube root of 8i =  

1c)

 

 

 

                    =

 

1d)

 

Recall,

 

 

 eπi(-i)

 

Question 2

Let

A = {z : 1 ≤ |z| ≤ 2} and B = {z : π/4 < Arg z < 3π/4}.

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

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

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

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

Question 3

In this question Γ is the line segment from i to 2

(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| = 1}

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) = z⁵-3z³+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

 -3z³+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 stream function for this flow. Hence find the equation of the streamline through the point 2i, and sketch this streamline, indicating 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 2i

(iii)  and  be the paths meeting at 2i and 0i given by

:  = 

 : ) = 

Sketch these paths, clearly indicating their directions

(iv) Using part (b)(ii), or otherwise, sketch the directions of g() and g() at g(2i).

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

Question 10

(a) Let f be the function

 

(i) Locate and classify the singularities of f.

(No justification required.)

(ii) Determine the Laurent series about 0 for f on the set

{z : 0 < |z| < 3},

giving the general term.

(iii) Determine the Laurent series about 3 for f on the set

{z : |z-3| > 3},

giving the general term.

(b) (i) Find the Taylor series about 0 (up to the term in z4) for the function

g(z) =  

and explain why the series represents g on C.

(ii) Hence evaluate the integral

,

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

QUESTION 11

(a) (i) Express  in Cartesian form, where z = x + iy, and hence show that

                  

      (ii) Determine  : −π ≤ Re z ≤ π, −1 ≤ Im z ≤ 1}, and find the point or points at which this maximum is attained.

(b) Show that the functions

                            f (|z| < 4)

and

                            g(z) =  (|z| > 4)

are indirect analytic continuations of each other.

Question 12

(a) Determine the extended Möbius transformation that maps

 i to 0,1 to ∞ and  to 1.

(b) Let

                            R = {z : |z| < 1, Re z + Im z > 1}

and

                            S = {w : Re w > 0, 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 f^-1.

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