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LANCASTER UNIVERSITY
2010 EXAMINATIONS
PART II (Third or Fourth Year) MATHEMATICS & STATISTICS 2 Hours Math 314: Integration
You should answer ALL Section A questions and TWO Section B questions. In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40. SECTION A
A1. (i) Show that the set^ Z[x] of polynomials that have integral coefficients is countable. You may assume that Zn^ is countable for each n ≥ 1. (ii) A real number is said to be algebraic, if it is the root of some non-zero polynomial with integral coefficients, and transcendental otherwise. Show that the set of algebraic numbers is countable, and hence that there exists a transcendental number. [12]
A2. (i) Define what is meant by a measurable function. (ii) For the function f (x) = sin(x^2 ), sketch a diagram to illustrate E = {x > 0 : f (x) > 12 } and determine E. Why is E measurable? [10]
A3. (i) State the Bounded Convergence Theorem. (ii) Show that the set S = {x ∈ (0, 1) : cos 1/x = ± 1 } is countable. (iii) Deduce that (^) ∫ (^1) 0 cos
n (^1) /x dx → 0 as n → ∞. [14]
A4. Evaluate the integral (^) ∫ (^1) 0 x
s− (^1) log x dx (s > 0). [8]
A5. State the main theorem (Σ) regarding the properties of Lebesgue measure. [6] please turn over
SECTION B
B1. (i) Define what is meant by an open subset of R. [3] (ii) Let E be a non-empty open subset of (a, b), and define ∼ on E by x ∼ y if [x, y] ⊆ E or [y, x] ⊆ E, for x, y ∈ E. Show that ∼ is an equivalence relation on E. [7] (iii) Hence show that E can be expressed uniquely as the union of a countable collection of mutually disjoint open intervals. [11] (iv) Show how to define the measure of E. [4] (v) Show that (0, ∞) ⊂
⋃^ ∞
n=
(n − 1 , n + 1) is an open cover that does not have a finite subcover. [5]
B2. (i) Define what is meant by an integrable function on (1,^ ∞).^ [2] (ii) Show that f (x) = cos x 2 x is an integrable function on (1, ∞) whereas g(x) = sin x^ xis not integrable. [Hint: Consider g on [nπ, (n + 1)π].] [14] (iii) By integrating by parts or otherwise, show that
Tlim →∞
∫ T
π/ 2
sin x x dx^ =^ −
π/ 2
cos x x^2 dx.^ [6] (iv) The Fourier cosine transform of h is ϕ(t) =
0 h(x) cos^ xt dx. Calculate the Fourier cosine transform of h(x) = I(0,1)(x), where I(0,1) denotes the indicator function of (0, 1). [4] (v) Plancherel’s formula asserts that 2 π
0 ϕ(t)
(^2) dt =^ ∫^ ∞ 0 h(x)
(^2) dx.
Apply this to the ϕ from (iv). [4]
please turn over
