A general technique is developed for calculating functional determinants of
second-order differential operators with Dirichlet, periodic, and antiperiodic
boundary conditions. As an example, we give simple formulas for a harmonic
oscillator with an arbitrary time-dependent frequency. Here our result is a
generalization of Gel'fand-Yaglom's famous formula which was restricted to
Dirichlet boundary conditions. Apart from the generalization, our derivation is
more transparent than theirs, the determinants requiring only knowledge of the
classical trajectories. Special properties of operators with a zero mode are
exhibited. Our technique does not require the calculation of the spectrum and
is as simple as Wronski's method for Green functions.