# Laplace Transform of Periodic Functions

## Abstract

By using the time shifting property and the series expansion of \(\frac {1}{1 - x}\) we arrive at a somehow magical equation furnishing the Laplace transform of an arbitrary periodic function. In particular assume that a non-periodic function has the Laplace transform *F*(*s*); then it follows that the periodic version of the function (with period *T*) has the Laplace transform \(\frac {F(s)}{1 - e^{-sT}}\). We apply this theorem to the periodic pulse and similar signals and illustrate the impact of being periodic on the spectrum. We notice that both real and imaginary parts of what used to be continuous smooth spectrum now transforms to being a jagged one with spikes overlapping the original envelop. The closer the period of the time signal, the more spiky the corresponding spectrum and the further away the spikes. On the other hand as we may the period of the time signal go to infinity the spikes in the frequency domain get closer and die out. Looking at it the other way around we are now also able to take an intrinsically periodic signal, such as the sine or cosine and use its transform to figure the spectrum of the aperiodic version of the signal! All theoretical steps and derivations are back by simulations and illustrative graphics.