= Green function of an undamped oscillator
{c}
{title2=$L=\partial_x^2+k^2$}
For zero initial value and slope, the causal <Green function> is $G_0(x,s)=H(x-s)\sin[k(x-s)]/k$, with $k\ne0$ and $H$ the <Heaviside step function>. For zero values at endpoints $a<b$, put $\kappa=|k|$. The <Dirichlet Green function> is
$$
G_D(x,s)=-\frac{\sin[\kappa(x_<-a)]\sin[\kappa(b-x_>)]}{\kappa\sin[\kappa(b-a)]}.
$$
Both are continuous at the source and have a first-derivative jump of one, giving $L_xG=\delta(x-s)$. The Dirichlet inverse exists when $\sin[\kappa(b-a)]\ne0$; a zero denominator signals a nontrivial homogeneous mode and a <Fredholm solvability condition>. The endpoint kernel is the zero-conjugation case of the <Dirichlet Green function for a conjugated harmonic oscillator>. Boundary conditions determine which kernel must be integrated against a forcing term.
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