Causal Fourier resolvent of a point-driven fixed string (${\hat{q}}_n(\omega )=\frac{2\sin (k_n{x}_0)}{L}\frac{\hat{F}(\omega )}{k_n^2-(\omega -i0{)}^2},k_n=n\pi /L$)

For $y_{tt}-y_{xx}=F(t)\delta (x-x_0)$ with zero endpoint displacement, a Fourier sine series gives $q_n^{\prime \prime }+k_n^2{q}_n=2F(t)\sin (k_n{x}_0)/L$. With the $e^{-i\omega t}$ Fourier transform convention, the causal inverse of this oscillator uses the displayed denominator. Its inverse transform is $H(t)\sin (k_{n}t)/k_n$, giving $q_n(t)=2\sin (k_n{x}_0)/(L{k}_n){\int }_{-\infty }^{t}F(s)\sin [k_n(t-s)]ds$ when the past integral exists. The plain denominator $k_n^2-{\omega }^2$ requires a pole prescription: its Cauchy principal value gives the time-symmetric kernel $\sin (k_n|t|)/(2k_n)$ instead. Homogeneous sine and cosine modes are additional freedom unless temporal data or causality are specified.