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PAPER TITLE: COMPLEX ANALYSIS

EXAM DATE: FRIDAY 3, JUNE 2016

COURSE CODE: M337/H

Question 1

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

(a)  

(b)

(c) the principal 6th root of −27

(d) i

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

1a)

 

Rationalize

 

 

 

 

1b)

 recall

 

So,

s  

 

Recall

 

Therefore,

 

 

                             =

Rationalize

1  

                                          =  

Recall

 

 

 

Therefore,

 

1c)

principal 6th root of −27 =

 

 

            =

 

 

Recall

 

Hence,

 

Therefore,

principal 6th root of −27 =

1d)

 

Recall

i=cosπ2+isinπ2  

i=eπ2i  

Therefore,

Note:

 

 

Question 2

Let

A = {z : |z| > 2} and B = {z : |Im z| ≥ 1}.

(a) Make separate sketches of the sets

A, B, C = B − A and D = A − B.

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

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

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

Question 3

In this question Γ is the circle {z : |z| = 3}

(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-3| = 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) =  + 4 .

(a) Determine the number of zeros of f that lie inside:

(i) the circle C₁ = {z : |z| = 3},

(ii) the circle C₂ = {z : |z| = 1}.

(b) Show that the equation iz5 + 4z2+2  = 0 has exactly 3 solutions in the set {z : 1 < |z| < 3}

Question 7

Let q(z) = 3z^--4i  be a velocity function.

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

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

(c) Evaluate the circulation of q along the path

Γ : γ(t) = t (t ∈ [0, 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) Determine whether or not lies in the Mandelbrot set M.

Question 9

(a) Use the Cauchy–Riemann Theorem and its converse to determine all the points of C at which the function

f(z) =  

is differentiable.

(b) Let g be the function .

(i) Show that g is conformal at 3 + i.

(ii) let  

:  = 2  cost + 2isin t (t ∈ [0, 2π])

 : ) =

Show that  and  intersect at the points 3+i, and Sketch  and   on the    same diagram

(iii) Describe the effect of g on a small disc centred at 3 + i and hence sketch, on a separate diagram, the approximate directions of the paths g( ) and g( ) near the point g(3 + i)

Question 10

(a) Let f be the function

 

(i) Locate and classify the singularities of f. 

(ii) Find the Laurent series about 1 for f on the annulus

{z : 1 < |z - 1| < 4},

giving the constant term and two terms on each side of it.

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

g(z) = cos(zexpz ),

and explain why the series represents g on C.

(ii) Hence evaluate the integral

,

where C is the circle {z : |z| = 2}.

Question 11

Let f be the function

 .

(a) (i) Find the residues of f 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) Determine the extended Möbius transformation   that maps

 1 to 0, ∞ to 1  and - 1 to ∞.

(b) Let

                            R = {z : |z + i| < √ 2}∩{z : |z − i| < √ 2},

                            S = {z₁ : 3π/4 < Arg_2π(z₁) < 5π/4},

                            T = {w : R_ew < 0}

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

(ii) Explain why  (R) = S.

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

(iv) Write down the rule of the inverse function .

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