HOMEWORK 7 — DUE NOV 7TH

MATH 25

There are two sections: A and B. Please return each part to the proper CA.

A. Problems graded by Benjamin

A.1. Prove that the function

f(x) = \begin{cases}0 & if x = 0 \\ x^{2} sin(1/x) & if x ≠ 0 \end{cases}

is differentiable everywhere on R but that f is not C^{1}.

A.2. Suppose that {f_{n}}_{n≥1} is a sequence of C^{1} functions f_{n} : [0, 1] → R, and that f, g : [0, 1] → R are two functions such that: for all x ∈ [0, 1], f_{n}(x) → f(x) (i.e. f_{n} → f pointwise), and f′_{n} → g uniformly on [0, 1].

Prove that f is C^{1}, that f′ = g and that f_{n} → f uniformly.

B. Problems graded by Inna

B.1. Let f : [0, 1] → R be a continuous function. Find the limit as n → ∞ of

I_{n} = n ∫_{0}^{1} x^{n}f(x)dx

and then find the next term in the asymptotic development (in order to do this, you can assume that f is as regular as you want, for example that f is C^{∞}).

B.2. For n ≥ 0, let J_{n} : R → R be the n-th Bessel function, defined by

J_{n}(z) = \frac{1}{π} ∫_{0}^{π} cos(nθ − z sin θ)dθ.

The following questions are largely independant:

(1) Explain why J_{n}(z) is a C^{∞} function of z.

(2) Prove that J_{1}(z) = −J′_{0}(z) and that J_{n−1}(z) − J_{n+1}(z) = 2J′_{n}(z).

(3) (extra credit) Prove that

J′′_{n}(z) + \frac{1}{z}J′_{n}(z) + (1 − \frac{n^{2}}{z^{2}} ) J_{n}(z) = 0.

(4) (extra credit) Prove that J_{n}(x) → 0 as x → +∞.