g(x) for all x, then R∞ 2 fn(x)dx → 0 as n

Real Analysis Homework #7

Due 11/17 1. Do the exercise given in class.

2. Let g(x) := 1/(x log x) for x > 1. Let fn = cn1A(n) for some constants c_n≥ 0 and measurable subsets A(n) of [2, ∞). Prove or disprove: If f_n(x) → 0 and |f_n(x)| ≤ g(x) for all x, then R∞

2 f_n(x)dx → 0 as n → ∞.

3. Let f (x, y) be a measurable function of two real variables having a partial derivative ∂f /∂x which is bounded for a < x < b and c ≤ y ≤ d, where c and d are finite and such that Rd

c |f (x, y)|dy < ∞ for some x ∈ (a, b). Prove that the integral is finite for all x ∈ (a, b) and that we can ”differentiate under the integral sign,” that is, (d/dx)Rd

c f (x, y)dy = Rd

c ∂f (x, y)/∂xdy for a < x < b.

4. (a) Show that R∞

0 sin(e^x)/(1 + nx²)dx → 0 as n → ∞.

(b) Show that R1

0(n cos x)/(1 + n²x^3/2)dx → 0 as n → ∞.

5. Show that if µ(X) < ∞, f_n → f in measure and g_n → g in measure, then f_ng_n → f g in measure. Does the statement still hold if µ(X) < ∞ is removed?

6. If m ≥ 0 is an integer, let J_m(x) =P∞ n=0

(−1)ⁿ

n!(n+m)!(x/2)^m+2n, Bessel func- tion of order m.

(i) Show that if a is a constant, 2R∞

0 J_m(2ax)x^m+1e^−x2dx = a^me^−a2. (ii) Show that if a > 1, R∞

0 J0(x)e^−axdx = (1 + a²)^−1/2.

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