Premium Resources

PAPER TITLE: COMPLEX ANALYSIS

EXAM DATE: MONDAY 8, JUNE 2015

COURSE CODE: M337/K

Question 1

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

(a)  

(b) 2 exp(3 − πi/3)

(c)

(d)

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

1a)

 

Rationalize

 

                      

                   =

Therefore,

 

1b)

 

 

 

 

Therefore,

 

         =

1c)

 

Recall

 

 

      =

 

 

 

    =

Hence,

 

1d)

 

 

And

 

 

 

            =

 

 

Question 2

Let

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

(a) Make separate sketches of the sets

A, B, C = A ∩ B and D = ∂A.

(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 line segment from i to 3 + i.

(a) Write down the standard parametrization for Γ.

(b) Evaluate

.

(c) Determine an upper estimate for the modulus of

 

Question 4

 Evaluate , where:

(a) C = {z : |z − 3| = 1}.

(b) C = {z : |z| = 1}.

(c) C = {z : |z + 3| = 1}.

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) =  + 5.

(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  + 5 has exactly 3 solutions in the set {z : 1 < |z| < 3}

Question 7

Let q(z) =   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 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) 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, but not analytic, at all the points on some line.

 

(b) Let g be the function .

(i) Show that g is conformal at 2i.

(ii) Describe the effect of g on a small disc centred at 2i.

(iii) Let  and  be the paths

:  =  

 : ) =

Show that  and  intersect at the points 2i, and and find the angle from  to  at this point of intersection

(iv) Sketch the paths  and  on the same diagram, clearly indicating their directions.

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

 (vi) By considering the effect of g on a suitable angle, show that g is not conformal at 0

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 : 2 < |z - 1| < 3},

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 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| = 3}.

(iii) Suppose h is an entire function that satisfies h(−z) = h(z), for all z ∈ C. Write down the value of

 

where C is the circle in part (b)(ii), and n is an even integer. Justify your answer briefly.

QUESTION 11

(a) Determine

                             

and find all points at which the maximum is attained, giving your answers in Cartesian form.

(b) Show that the functions

                             (|z + 1| < 1)

and

                            g(z) =  (|z + i| < 1)

are direct analytic continuations of each other.

Question 12

(a) Determine the extended Möbius transformation that maps

 1 to 0,1 to ∞  and ∞ to 1-i.

(b) Let

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

                            S = {z₁ : π/2< Arg_2π(z₁) < π},

                            T = {w : Imw < 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 .

(v) Is f the only conformal mapping from R onto T? Briefly justify your answer.

 

Last updated: Sep 02, 2021 02:14 PM

add to cart ← Back