We know the secret of your success
PAPER TITLE: COMPLEX ANALYSIS
EXAM DATE: TUESDAY 11, OCTOBER 2011
COURSE CODE: M337/C
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)
Therefore,
=
1b)
e
1c)
1d)
Question 2
Let
A = {z : -1and B = {z : 2<| < 3}.
(a) Make separate sketches of the three sets A, B and B − A.
(b) For each of the sets A, B and B − A, write down whether or not the set is:
(i) open
(ii) connected
(iii) a region
(iv) bounded
(v) compact.
Question 3
(a) Evaluate
.
Γ is the line segment from i to 1.
(b) Determine an upper estimate for the modulus of
where C is the circle {z : |z| = 2}
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
a)
b)
c)
Question 5
(a) Find the residues of the function
at each of the poles of f.
(b) Hence evaluate the integrals
-
Question 6
This question concerns the solutions of the equation.
(a) Use Rouch´e’s Theorem to show that there are exactly two solutions in the annulus {z : 1 < |z| < 2}.
(b) Determine how many solutions lie in the upper half-plane {z : Im z > 0}.
Question 7
Let q(z) = be a velocity function.
(a) Explain why q represents a model fluid flow on C – {0}.
(b) Determine a stream function for this flow. Hence find the equation of the streamline through the point 2, and sketch this streamline, indicating the direction of flow.
(c) Evaluate the circulation of q along the path
Γ : γ(t) = 2t (t ∈ [1, 2]).
Question 8
(a) Show that the iteration sequence
is conjugate to the iteration sequence
= , n = 0, 1, 2,...,
(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 determine the set of points S at which f is differentiable.
(iii) Show that the derived function f is constant on S.
(b) Let g be the function .
(i) Show that g is conformal on C − {0}.
(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).
Question 10
(a) Write down the domain A of f.
(b) Show that the Laurent series about 0 for f is
Hence evaluate
where C is the unit circle {z : |z| = 1}.
(c) Use the Uniqueness Theorem to show that f is the only analytic function with domain A such that
(d) Show that f has singularities at points of the form 2kπ, k ∈ Z, and classify these singularities. (Hint: cos z = cos(z − 2kπ) for k ∈ Z.)
QUESTION 11
(a) Show that the functions
(|z| < 3)
and
g(z) = (|z| > 3)
are indirect analytic continuations of each other.
(b) Determine
and find all points at which the maximum is attained, giving your answers in Cartesian form.
Question 12
(a) Determine the extended Möbius transformation that maps
-1-i to 0, 0 to 1 and 1+i to ∞
(b) Let
R = {z : 3π4< Imz< 7π4},
S = {z1 : Re z1 + Im z1 < 0},
T = {w : |w| < 1}
(i) Sketch the regions R, S and T on separate diagram.
(ii) Explain why (S) = T, where is the extended M¨obius transformation from part (a).
(iii) Hence determine a one-one conformal mapping f from R onto T. (Hint: you may find the exponential function useful.)
(iv) Obtain a rule for the inverse function .
(v) Hence find the point p in R that maps to 0. State, with a reason, whether every conformal mapping from R to T maps p to 0?
(Purchase full paper by adding to cart)
Last updated: Sep 02, 2021 02:13 PM
Your one-stop website for academic resources, tutoring, writing, editing, study abroad application, cv writing & proofreading needs.
