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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 . 1 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 2