Math205b Winter Term 2003

Assignment 10

1. Let φ ∈ C(ℝ, ℝ) and f ∈ C([a, b], ℝ) for a < b ∈ ℝ. Assume that φ is convex, that is, that it satisfies

φ((1 − t)x + ty) ≤ (1 − t)φ(x) + tφ(y) , x, y ∈ ℝ , t ∈ [0, 1] .

Prove the validity of the following Jensen’s inequality

φ(\frac{1}{b − a} ∫_{a}^{b} f(x) dx) ≤ \frac{1}{b − a} ∫_{a}^{b} φ(f(x)) dx .

2. The Legendre ploynomials L_{n} can be defined by

P_{n}(x) = \frac{1}{2^{n} n!} \frac{d^{n}}{dx^{n}} (x^{2} − 1)^{n} , n = 0, 1, 2, …

Show that

∫_{−1}^{1} P_{n}(x)P_{m}(x) dx = 0 , m ≠ n .

3. Show that the improper integral

Γ(α) = ∫_{0}^{∞} t^{α}e^{−t} dt

exists for every ℝ ∋ α > −1 and verify that

Γ(n) = n! , n ∈ ℕ .

4. Assume that f ∈ C^{1}([−1, 1]) and prove that the following principal value integral exists (i.e. that the defining limit always exists)

p.v. ∫_{−1}^{1} \frac{f(x)}{x} dx = lim_{ε→0} ∫_{|x|≥ε} \frac{f(x)}{x} dx .

Also show that

∫_{−1}^{1} log(|x|)f′(x) dx = −p.v. ∫_{−1}^{1} \frac{f(x)}{x} dx .

5. You ask a question.

The Homework is due Friday January 24