Topics in Analysis Examples Sheet 3 W. T. G.

1. For each n, let f_{n} : [0, 1] → ℝ be a function and suppose that the functions f_{n} converge uniformly to a function f. Suppose also that f is bounded (above and below). Prove that for any positive integer m the functions g_{n}(t) = f_{n}(t)^{m} converge uniformly to g(t) = f(t)^{m}.

2. Prove that there is no sequence of analytic functions f_{n} that converges uniformly on the unit circle to the function 1/z (which on the circle is the same as z̄). Why does this not contradict Runge’s theorem?

3. Construct a sequence of polynomials that converges uniformly to 1/z on the semicircle consisting of all points of the unit circle that have real part greater than or equal to 0.

4. Work out continued-fraction expansions for 71/49 and \sqrt{3}.

5. Define a sequence (x_{n}) by x_{1} = 1, x_{2} = 3 and x_{n} = x_{n−1} + x_{n−2} for n ⩾ 3. Prove that x_{n}/x_{n−1} converges to the golden ratio (1 + \sqrt{5})/2. Describe the continued-fraction expansion of x_{n}/x_{n−1}.

6. Let x have continued-fraction expansion

1 + \frac{1}{2 + _{\frac{1}{1+_{\frac{1}{2+ \frac{1}{...}}}}}}

with the 1s and 2s continuing to alternate. Calculate the value of x.

7. What rational with denominator less than 10 best approximates the number 71/49?

8. Let C be a circle with centre 0 in the complex plane, traversed anticlockwise, and let K be a compact set disjoint from C. Define a function on K by

f(z) = \frac{1}{2πi} ∫_{C} \frac{dw}{w − z} .

By splitting up the path integral into small pieces and approximating the contribution from each piece, prove that f can be uniformly approximated on K by functions of the form f_{n}(z) = ∑_{i=1}^{N} a_{i}/(w_{i} − z), where the a_{i} and w_{i} are complex numbers with the w_{i} lying in C.

9. Prove that \sqrt{3} + \sqrt{5} and e^{2} are irrational.