Math 295a Fall Term 2001

Assignment 1

1. Show that v.p.\frac{1}{x} is a well-defined distribution. Recall that

⟨v.p.\frac{1}{x}, φ⟩ = lim_{ε→0} ∫_{|x|≥ε} \frac{φ(x)}{x} dx

for any test function φ ∈ D(ℝ). Then prove that

u_{ε}^{±} → ∓iπδ + v.p.\frac{1}{x} as ε ↓ 0

for u_{ε}^{±}(x) := \frac{1}{x±iε} , x ∈ ℝ, in the sense of distributions.

2. Compute f′ and f′′ for f(x) = log |x| , x ∈ ℝ and f(x) = |x| , x ∈ ℝ, respectively, in the sense of distributions.

3. Let ρ : ℝ ↦ ℝ be an integrable function with

supp(ρ) ⊂ [0, 1] and ∫_{−∞}^{∞} ρ(x) dx = 1 .

Show that ρ_{ε} → δ in D′(ℝ) as ε → 0 for ρ_{ε}(x) := \frac{1}{ε}ρ(εx) , x ∈ ℝ.

4. Consider the function u defined as

u(x, y) = \frac{1}{π} \frac{y}{x^{2} + y^{2}} , (x, y) ∈ ℝ × (0, ∞)

Then u ∈ C^{∞}(ℝ × (0, ∞)). Compute lim_{y→0} u(·, y).

5. Assume that f ∈ C^{1}(ℝ^{n}, ℝ), then

T_{∂^{j}f} = ∂^{j}T_{f} , j = 1, …, n

that is, the derivative in the sense of distributions coincides with the classical derivative if the latter exists. Let now f be continuously differentiable except at finitely many points x_{1}, …, x_{m} (m ∈ ℕ) where

lim_{x↓x_{j}} f(x) − lim_{x↑x_{j}} f(x) = σ_{j} , j = 1, …, m .

Compute its derivative in the sense of distributions.