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PAPER TITLE: COMPLEX ANALYSIS

EXAM DATE: TUESDAY 14, OCTOBER 2003

COURSE CODE: M337/C

Question 1

Give the cartesian form of the following complex numbers, simplifying your answers as far as possible.

(a)  

(b) the square root of  

(c) l

(d)

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

1a)

 

 

 

1b)

 

Recall,

 

 

 

= 

 

          =

 

1c)

 

 

 

1d)

 

 

 

 

Therefore,

 

 

Question 2

Let

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

(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 bounded

Question 3

In this question Γ is the line segment from 1to 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| = 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 all its poles.

(b) Hence evaluate the integrals

                           

Question 6

(a) Evaluate the following expression where Γ is a gamma function

(i) Γ(5)

(ii) Γ(5/2)

(iii)

(b) Prove that the series

 

Converges uniformly on a set E = {z:0

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

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) By considering the sequence , show that 1 does not belong to the Mandelbrot set M. Hence, using the Fatou-Julia theorem . determine whether or not 0 is in the keep set  

Question 9

(a) Show that the  function  is not differentiable at 0 by using the following method.

(i) directly from the definition of the derivative as a limit

(ii) by using the Cauchy-Riemann theorem.

(b) Let g be the function g(z) =

(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

 

Write down the singularities of f and determine their nature.

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

 

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

h(z) = Log(cosh z).

 Hence determine the first three non-zero terms of the Taylor series about 0 for tanh.

Question 11

Let  be the function

.

 Show that

 i) f has no zeros in {z : |z| ≤ 2},  

ii) f has no zeros on the real axis

iii) f has exactly 3 zeros in {z : |z| > 2, Imz >0},  

(iv) f has exactly one  zero in each of the 4 -regions bounded by the real and imaginary axes , given that f has exactly two zeros on the imaginary axis

b) Evaluate the improper real integral

          

Question 12

In this question g is the Möbius transformation given by

 

a) i) the point 1 and β are inverse point with respect to the extended imaginary axis. Write down the value of β and the image of 1 and β, under g

ii)Deduce that the image of the extended imaginary axis under g is the unit circle {w: |w| =1}

iii) Hence or otherwise show that image of the region {z: Rez > 0} under g is D = {w:|w| < 1},

b) i) Sketch the region

     R = {z : |z | < 1, Rez  < 0}, and  = {z₁ :  

ii) Determine a Möbius transformation  which maps R to  (you must justify that  maps the region correctly)

iii) using  and g defined above and any intermediate conformal mapping needed. Determine the rule of conformal mapping f from R to D ={w:|w| < 1}. You do not need to simplify or justify your answer that it is conformal

c) Write down an example of each of the following

i) a Möbius transformation which is conformal at C

ii) a function which is conformal at C but not a Möbius transformation

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