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PAPER TITLE: COMPLEX ANALYSIS

EXAM DATE: THURSDAY 13, JUNE 2019

COURSE CODE: M337/G

Question 1

Let w = −1 +  .

(a) Determine each of the following complex numbers, giving your answers in Cartesian form and simplifying them as far as possible.

(i) 1/w^-

(ii) Log w

(iii) Log(w4)

(b) Find all the cube roots of w in polar form.

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

1ai)

1/w^-

 

1/w^-  =

Rationalize

 

 

1/w^-  =

1aii)

Log w = ln|w|+iArgw

 

        =  

       =

 

 

 

recall,

Log w =

Log w =

1aiii)

Log(w4)

 

( 

Log() = Log()

Log() =  

 

         = 

         =

 

 

 

 

Recall,

Log() =

Log() =

1b)

 

 

 

 

w =

w in polar from

w

w =   

 

 

 

Therefore,

 

Question 2

(a) Evaluate the following integrals with C = {z : |z| = 1}. Name any standard results that you use and check that their hypotheses are satisfied.

i)

ii)

iii)

(b) Explain how the three answers you obtained in part (a) would change if C was replaced by the contour Γ : γ(t) = (t ∈ [0, 2π])

Question 3

Consider the equation + 4z + 2i = 0.

(a) Use Rouch´e’s Theorem to determine the number of solutions of the equation in the annulus {z : 1 < |z| < 2}.

(b) Determine the number of solutions of the equation with modulus at least 2.

Question 4

Let f be the Möbius transformation

 

 and let C = {z : |z − i| = 1}.

(a) Find the point β such that α = ∞ and β are inverse points with respect to C.

(b) Determine an equation for the image circle f(C) in Apollonian form.

(c) Find the centre and radius of f(C)

Question 5

Let q be the velocity function

q(z) = .

(a) Explain why q is the velocity function for an ideal flow on C − {0}.

(b) Determine a stream function for this flow, and hence find an equation for the streamline through the point 2.

(c) Sketch this streamline and indicate the direction of flow.

(d) Find the flux of q across the line segment Γ from −1 + i to 1 + i.

Question 6

(a) Show that the point α = is a periodic point of the function

f(z) =  .

 Find the multiplier of the corresponding cycle, and determine whether the cycle is attracting, repelling or indifferent.

(b) Determine whether the point −1 − i  belongs to the Mandelbrot set.

(c) Find a real number c in the Mandelbrot set such that −c does not belong to the Mandelbrot set.

Question 7

(a) Use the Cauchy–Riemann Theorem and its converse to determine all the points

z = x + iy of C at which the function

f(x + iy) = sin(y + ix)

is differentiable.

(b) Let  and  be the smooth paths

 :  = t +1+ it (t ∈ R),

 : γ) = 2 +(t ∈ [0, 2π]).

(i) Show that  and  meet at the point 1 and find the angle from  to  at 1.

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

(iii) Let g(z) = Log z. Find the angle from the path g() to the path g() at the point g(1).

Question 8

(a) Find the Taylor series about 0 for the function

f(z) =  

up to the term in .

(b) Find the Laurent series about 1 for the function

g(z) =

on the annulus {z : 1 < |z − 1| < 3}, giving the constant term and two terms on each side of it.

(c) Let  and  be entire functions such that

(x) = exp((x)), for all x ∈ R.

Prove that

(z) = exp((z)), for all z ∈ C.

Question 9

Let f be the function

 .

(a) (i) Find the residues of f at each of the points 0, i and .

(ii) Hence determine the sum of the series

 

(iii) Use your solution to part (a)(ii) to prove that

           

(b) Find all the zeros of f and prove that f is one-to-one near each of these zeros.

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