Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 8 / b / iv

Split the inverse transform into the intervals $|\omega |<\mu$ and $|\omega |>\mu$, and in each case pair the positive-frequency integral with its complex conjugate at negative frequency. The low-frequency pair becomessince $B=0$ and the decaying spatial factor is real. In the high-frequency positive half, replace $A(\omega )$ by $B(-\omega {)}^{\ast }$. The negative half is again its conjugate. Combining the two pieces givesEvery integrand satisfies the massive wave equation because ${\omega }^2=k^2+{\mu }^2$. Linearity, and taking the real part, preserve the equation whenever differentiation under the integrals is justified, or distributionally under the corresponding spectral assumptions. The construction is exactly the real bounded Fourier modes of the Klein-Gordon equation decomposition.

The displayed expression uses ordinary spectral amplitudes. If distributions supported at $\omega =\pm \mu$ are admitted, a further bounded threshold contribution $c_1\cos \mu t+c_2\sin \mu t$, independent of $r$, is possible. It is not a missing propagating or evanescent branch of the open frequency intervals.