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PAPER TITLE: COMPLEX ANALYSIS

EXAM DATE: FRIDAY 9, JUNE 2017

COURSE CODE: M337/J

Question 1

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

 

 

 

 

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

1a)

 

 

 

 

 

           =

Therefore,

 

1b)

 

 

 

  

   =

1c)

 

Rationalize

 

 

                         =

Modulus =

               =

Arg(=

                 =

Arg( 

Therefore,

 

Question 2

Let A = {z : 0 < Re z < 2} and B = {z : 1 ≤ |z| ≤ 2}.

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

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

(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.

(c) Give an example of a set in C that is none of the following: open, closed, bounded, connected.

Question 3

Let Γ be the line segment from 1 to −i.

(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| = 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

 and .

Question 6

Let f(z) = 3 + .

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

(i) the circle C₁ = {z : |z| = 1},

(ii) the circle C₂ = {z : |z| = 2}.

(b) Show that the equation 3 + . = 0 has at least one real solution.

Question 7

Let q(z) = 3/be a velocity function.

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

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

(c) Evaluate the flux of q across the circle {z : |z| = 2}.

Question 8

(a) Find the fixed points of the function f(z) =  and classify them as attracting, repelling or indifferent.

(b) Determine which of the following points c lie in the Mandelbrot set:

(i) c = −1 − i,

(ii) c =.

Justify your answer in each case.

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  and  be the paths

 :  =  (t ∈ [0,2π]),

 : ) =  (t ∈ R).

(i) Show that  and  intersect at the points −1 and −i and find the angle from  to  at each point of intersection.

(ii) Sketch  and  on the same diagram,

(iii) Let g be the function

g(z)= 

Show that g is conformal at −i.

(iv) Determine the angle from the path g() to the path g() at the point g(−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) = exp(sinh 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) Determine

                             

and find all points at which the maximum is attained.

(b) Show that the functions

                             (|z | < 7)

and

                            g(z) =  (|z | > 7)

are indirect analytic continuations of each other

Question 12

(a) Determine the extended Möbius transformation  that maps

 i to 0,  and - i to ∞.

(b) Let

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

and

                            S = {}, .

(i) Sketch the R, S .

(ii) Determine the region   (R) and Sketch this region.

((iii) Hence determine a one-one conformal mapping g from R onto S.

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

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