= Causal Fourier resolvent of a point-driven fixed string
{title2=$\widehat q_n(\omega)=\frac{2\sin(k_nx_0)}L\,\frac{\widehat F(\omega)}{k_n^2-(\omega-i0)^2},\quad k_n=n\pi/L$}
For $y_{tt}-y_{xx}=F(t)\delta(x-x_0)$ with zero endpoint displacement, a <Fourier sine series> gives $q_n''+k_n^2q_n=2F(t)\sin(k_nx_0)/L$. With the $e^{-i\omega t}$ <Fourier transform> convention, the causal inverse of this oscillator uses the displayed denominator. Its inverse transform is $H(t)\sin(k_nt)/k_n$, giving $q_n(t)=2\sin(k_nx_0)/(Lk_n)\int_{-\infty}^tF(s)\sin[k_n(t-s)]ds$ 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.
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