5. Laplace Transform of a Periodic Function f(t)

If function f(t) is:

Periodic with period p > 0, so that f(t + p) = f(t), and

f1(t) is one period (i.e. one cycle) of the function, written using Unit Step functions,

then

`Lap{f(t)}= Lap{f_1(t)}xx 1/(1-e^(-sp))`

NOTE: In English, the formula says:

The Laplace Transform of the periodic function f(t) with period p, equals the Laplace Transform of one cycle of the function, divided by `(1-e^(-sp))`.

Examples

Find the Laplace transforms of the periodic functions shown below:

(a)

Graph of periodic unit ramp function.

(b) Saw-tooth waveform:

Graph of saw-tooth waveform.

(c) Full-wave rectification of sin t:

Graph of `f(t)=sin t*{u(t)-u(t-pi)}`, with period `pi`.