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PAPER TITLE: COMPLEX ANALYSIS

EXAM DATE: THURSDAY 7, JUNE 2018

COURSE CODE: M337/E

Question 1

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

(a)

(b)  

(c)  

(d)

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

1a)

 

  

1b)

 

 

 

 

Rationalize

- 

 

 

1c)

 

       =

       =

 

 

 

1d)

 

Recall 

 

( 

 

            =

            =

 

Question 2

Let A = {z : 0 < |z − i| < 3} and B = {z : 0 ≤ Im z ≤ 3}.

(a) Make separate sketches of the sets A, B, C = A ∪ ∂A and D = B − A. There is no need to calculate any of the boundary points of D.

(b) For each of the sets A, B, C and D, write down whether the set is

(i) compact,

(ii) a region.

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

                            C1 = {z : |z| = 1}, C2 = {z : |z| = 2} and C3 = {z : |z| = 3}.

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

f(z) = at each of its poles.

(b) Hence evaluate the improper integral

Question 6

Use Rouch´e’s Theorem to determine the number of solutions of the equation  in the annulus {z : 1 < |z| < 3}

Question 7

Let q(z)=4iz¯ 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 1 − i and indicate the direction of flow.

(c) Evaluate the flux of q across the path Γ : γ(t) = t (t ∈ [0, 1])

Question 8

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

(b) Determine whether or not the points and 1 + i  lie in the Mandelbrot set.

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

 :  = 1+ it (t ∈ R),

 : ) =  + (t ∈ R).

(i) Show that  and  intersect at the point

(ii) Sketch  and  on the same diagram, indicating the directions of increasing values of t.

(iii) Let g(z) = . Explain why the angle from to  at the point  is equal to the angle from the path g() to the path g() at the point g().

Question 10

(a) Let f be the function

                           .

Find the Laurent series about i for f on the punctured disc

{z : 0 < |z − i| < 1},

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

(b) Let

                            g(z) = cosh(sinh(1/z)).

(i) Find the Laurent series about 0 for g, giving the constant term and all terms as far as the term in .

ii) Hence evaluate the integral

                           

where C is the circle {z : |z| = 1}.

(iii) Calculate g(2/(nπi)), for n = 1, 2,.... Hence, or otherwise, deduce that g has an essential singularity at 0.

QUESTION 11

(a) Determine

                            max{| exp(iz5)| : |z| ≤ 2}, 

and find all points at which the maximum is attained.

(b) Show that the functions

                             (|z + 1| < 2)

and

                            g(z) =  (|z + 1| > 2)

are indirect analytic continuations of each other

Question 12

(a) Determine the extended Möbius transformation  that maps

 0 to 0, and 1 + i to ∞.

(b) Let

                            R = {z : |z − 1| < 1, |z − i| < 1},

                            S = {z : −π/4 < Arg z < π/4},

                            T = {z : Re z > 0}.

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

(ii) Show that  (R) = S.

(iii) Determine a one-one conformal mapping from S onto T .

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

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