Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 5 / b / Solution

Apply the Fourier transform of a derivative twice to the damped harmonic oscillator equation. Under the corresponding convergence assumptions,ThusSince $\kappa ,\Omega >0$, this denominator has no real zero: its imaginary part could vanish only at $\omega =0$, where its real part is positive. The transform therefore defines a regular response on the real frequency axis.