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PAPER TITLE: COMPLEX ANALYSIS

EXAM DATE: TUESDAY 8, OCTOBER 2002

COURSE CODE: M337/D

Question 1

Determine each of the following complex numbers in cartesian form.

(a)  

(b)

(c) The square root of

(d)

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

1a)

 

 

            =

            =

 

1b)

 

 

 

 

 

 

1c)

 

Recall

 

 

 =

 

 

 

1d)

 

Recall

 

 

 

Question 2

Let

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

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

(b) write down which of the set A, B and C is a region and for any which are not region explain why not

(c) For each region you have found in part (b) state whether or not it is simply connected.

(d)  Sketch a simple closed contour Γ in one of the set A,B or C for which

 

Question 3

(a) Evaluate

.

Where C is the circle {z : |z| = 2}

(b) Determine an upper estimate for the modulus of

 

Question 4

Let  be the function  Evaluate  where

a) C = {z : |z| = }

b) C = {z : |z| = 2}

Question 5

(a) Find the residues of the function

 

at all its poles.

(b) Hence evaluate the integrals

 

Question 6

Consider the following function

 

1. i) explain why  on the half plane 

ii) Hence prove that  are direct analytic continuation of each other

b) prove that

           

Form a closed chain but

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) Why is 0 neither a source or a vortex

Question 8

(a) Prove that the function  has two fixed point and determine their nature.

(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)Hence determine all point of C  at which function f is differentiable.

(b) Let g be the function .

(i) Show that g is conformal at.

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

:    =2t  

 : ) =

Show that   meet at point 1and  Sketch  on the same diagram,

(iii) Describe the effect of g on a small disc centred at 1 And hence make a sketch showing the approximate directions of the path g( ) and g( ) near the point g(1).

Question 10

(a) Let f be the function

 

(i) Locate and classify the singularities of f.

(ii) Determine the Laurent series about 1 for f on  

Giving an expression for the general term.

(b) (i) Find the Taylor series about 0 (up to the term in ) 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) Show that if |z| = 2 then 

       ii) Use Rouch’e’s theorem to determine how many solution of the equation    lie inside the circle C₂ = {z : 1<|z| <2  1

(b) Evaluate the improper real integral

 

Question 12

a)  State whether each of the following assertion is true or false, briefly justify your answer.

 i) The reciprocal function is a Möbius transformation

ii) There is a unique ono-one conformal mapping from the half plane {z:Rez>0}  onto the unit disc {w:|w| <1}

iii) There is a Möbius transformation from C onto the unit disc {w:|w| <1}

b) let f be the Möbius transformation

                           

i) Prove that f maps the half plane {z:Rez>0}  onto the unit disc {w:|w| <1}

ii) Find the image under the function f of the real axis

iii) by composing the function f of part b(i) with an appropriate mapping obtain a formula for a one-one conformal mapping g from the region C {x∈R:x≤0}  onto D

iv) write down a formula for the corresponding inverse function

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