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PAPER TITLE: COMPLEX ANALYSIS

EXAM DATE: FRIDAY 15, OCTOBER 2004

COURSE CODE: M337/K

Question 1

(a) Write down the values of

(i)  

(ii) Arg α

(b) hence or otherwise evaluate each of the following  giving your answers in Cartesian form.

(i)  

(ii)

(iii)

(iv)

ANSWERS(Purchase full paper to get all the solutions):

1ai)

 

 

 

 

1aii)

Arg

            =

Arg

1bi)

α 

 

Rationalize

 

 

1bii)

 

                             =

 

1biii)

 

 

1biv)

 

 

 

 

                    =

 

 

Question 2

Let

A = {z : 1 < |z+i| < 2} and B = {z : 0 ≤ Arg z ≤ π/2}.

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

(b) write down which of the set A, B and C = A – B, if any is  

(i) a region

(ii) a simply- connected region

(iii) neither open nor closed

(c) using set notation, give an example of a set which is closed but not connected

Question 3

In this question Γ is the circle {z : |z| = 2}

(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-i| = 2}

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) = .

(a) i) Show that f has three zeros lying inside the circle C₁ = {z : |z| = 2},

       ii) Determine the number of zeros of f which lies inside the circle C₂ = {z : |z| = 1}.

(b) Show that f has a zero inside C₂ which is real and positive

Question 7

Let q(z) =  be a velocity function on C.

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

(b) Determine a stream function for this flow. Hence find the equation of the  the streamline through the point i and the streamline through the point 1 + i. And sketch the streamline indicating the direction of flow.

(c) Determine the flux of q across the path  where.

 

Question 8

(a) Show that the iteration sequence

        

 is conjugate to the iteration sequence

        =  , n = 0, 1, 2,...,

with .

(b) Find the fixed points of  and determine their nature.

(c) Determine whether or not lies in the Mandelbrot set M.

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 2.

(iii) and  are smooth paths meeting at 0 and 2 given by

:  =2t   

 : ) =

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(2).

(v) Show that g is not conformal at 0.

Question 10

(a) Let f be the function

 

(i) Write down the singularity and determine their nature.

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

{z : 0 < |z| < 1},

giving the general term.

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

{z : |z-i| > 1},

giving the general term.

(b)(i)  Determine the Laurent series about 0  for the function g defined by

g(z) =   giving the non-three vanishing term.

(ii) Classify the singularity of g at 0, justifying your answer.

(iii) show that

 ,

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

(iv) write down the value of

 

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

Question 11

(a) Find the residues of function

 .

 at each of the points 0,  and .

(b) Hence determine the sum of the series

 

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

           

Question 12

(a) (i) show that  are inverse point with respect to the circle C = {z:|z|=2}

        (ii) find the image of   under the Möbius transformation

                           

and hence sketch the image of C under g

        (iii) indicate g(D) on your sketch when D = {z:|z|<2}

 

(b) Let

                            R = {z : || < 1, 0z -1) < π},

                             = {z₁ :

                            S = {w : Imw > 0}

(i) Sketch the regions R,  and S

(ii) Find a Möbius transformation  which maps R to  and use the conformality of  to justify that it does not indeed map R to

((iii) Write down a conformal mapping from  to S and hence a conformal mapping f from R to S.

(iv) Explain why the function f is not conformal on R? (the closure of R).

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