  We derive an exact expression for the transmission amplitude of a particle
moving through a harmonically driven delta-function potential by using the
method of continued-fractions within the framework of Floquet theory. We prove
that the transmission through this potential as a function of the incident
energy presents at most two real zeros, that its poles occur at energies
$n\hbar\omega+\varepsilon^*$ ($0<Re(\varepsilon^*)<\hbar\omega$), and that the
poles and zeros in the transmission amplitude come in pairs with the distance
between the zeros and the poles (and their residue) decreasing with increasing
energy of the incident particle. We also show the existence of non-resonant
"bands" in the transmission amplitude as a function of the strength of the
potential and the driving frequency.
