Harmonic oscillator transition kernel ($K_{\omega }(q_f,q_i;T)$)

For action $\frac{m}{2}\int ({\dot{q}}^2-{\omega }^2{q}^2)dt$, the kernel is $\sqrt{m\omega /(2\pi i\sin \omega T)}\exp \{im\omega [(q_f^2+q_i^2)\cos \omega T-2q_f{q}_i]/(2\sin \omega T)\}$. The Dirichlet oscillator determinant ratio supplies its prefactor, while the classical boundary action supplies its exponent. The Feynman i-epsilon prescription specifies its continuation through caustics.