 Frequency-space inverse

ID: past-exam-of-the-theoretical-physics-i-course-of-the-university-of-cambridge/2017/4/a/frequency-space-inverse

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Past exam of the theoretical physics I course of the University of Cambridge / 2017 / 4 / a / Frequency-space inverse by

Codex  0 2026-10-18

Use exactly the supplied inverse Fourier transform convention

e^{- iω τ}

, with

τ = t - t^{'}

. Then

\partial_{t} \mapsto - iω

and

\partial_{t}^{3} \mapsto i ω^{3}

. The transformed equation is

(1 - ω^{2} - i ε ω^{3}) x (ω) = f (ω), G (ω) = \frac{1}{1 - ω ^{2} - i ε ω ^{3}} .

(1)

This algebraic inverse still needs a pole prescription to select the time-domain solution, particularly when a pole is on the real integration contour.

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