 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2/5/a/iv/solution

Multiply

H

f (ω) = \frac{1}{2 i} [H (ω - ω_{0}) - H (ω + ω_{0})] .

Inserting part (iii) gives

f (ω) = \frac{1}{2} [PV \frac{1}{ω + ω _{0}} - PV \frac{1}{ω - ω _{0}}] + \frac{π}{2 i} [δ (ω - ω_{0}) - δ (ω + ω_{0})] .

The reciprocal terms are real principal-value distributions and the delta terms are imaginary. Equivalently, exponential damping

H (t) e^{- ϵ t}

gives

ω_{0} / [(ϵ + iω)^{2} + ω_{0}^{2}]

, whose distributional limit is the boxed expression. The shifted-difference formula remains valid at

ω_{0} = 0

, where all terms cancel.

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