Download Assignment 9 (for submission in the week beginning 5

MTH4110/MTH4210
Mathematical Structures
Assignment 9
For submission in week 11
Questions marked with an asterisk (∗) are to be handed in to your tutor during the tutorial in the week indicated. They will be marked and
returned to you in the following tutorial.
Questions marked with a dagger (†) are for you to think about before the
tutorial. You can discuss them with the tutor during the tutorial.
Other questions (those marked “Project”) are optional. If you want feedback on them, please hand them in to the lecturer within four weeks of the
assignment being due. A prize may be offered for good work on project
questions.
1 (∗) Let f : C → C be given by
f (z) = z2 + i .
(a) Is f a function?
(b) Is it injective?
(c) Is it surjective?
2 (†) Find the real and imaginary part of
1+
1+ 1+i
1−i
1− 1+i
1−i
1−
1+ 1+i
1−i
1− 1+i
1−i
.
3 (†) Use De Moivre’s Theorem to express cos 4θ as a polynomial in cos θ , and to
express sin 4θ as cos θ multiplied by a polynomial in sin θ .
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4 (†) One of the following is not a function; which one, and why? In the other cases,
is the function one-to-one? is it onto?
(a) The function F : C → R given by F(z) = arg(z).
(b) The function F : R → R given by F(x) = x3 − x.
(c) The function F : R → R given by F(x) = ex .
5 (†) Is this proof valid?
Theorem cos2 x + sin2 x = 1.
Proof Let f (x) = cos2 x + sin2 x.
Differentiating, f 0 (x) = 2 cos x(− sin x) + 2 sin x cos x = 0. Hence f (x) is constant.
But f (0) = 1; so f (x) = 1 for all x.
6 (From the summer exam 2016) Let F : C → C be a function. A complex number w
is said to be a fixed point of F, if F(w) = w.
(a) Show that 2 − i is a fixed point of the function f given by f (z) = z2 − 3z + 5.
(b) Does every polynomial of degree greater 2 have a fixed point? Give reasons for
your answer.
7 (Project) Let r be an irrational real number greater than 1. The Beatty sequence Br
is the sequence of natural
√ numbers whose nth term is bnrc.
For example, if r = 2 = 1.41421 . . ., then the terms of Br are
b1.41421 . . .c = 1, b2.82843 . . .c = 2, b4.24264 . . .c = 4, b5.65685 . . .c = 5,
b7.07107 . . .c = 7, b8.48528 . . .c = 8, b9.89949 . . .c = 9, . . .
The gaps in the sequence, namely 3, 6, 10, . . ., form another interesting sequence.
(a) Rayleigh’s Theorem asserts that the gaps in the Beatty sequence Br form the
Beatty sequence Bs , where 1/r + 1/s = 1. Find a proof of this theorem and
express it in your own words.
√
(b) In the case where r = 2, let pn be the nth term in the sequence Br , and qn the
nth term in the sequence of gaps. Prove that qn = pn + 2n.
OFB 30/11/16
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