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

If function *f*(*t*) is:

Periodicwith periodp> 0, so thatf(t+p) =f(t), and

f_{1}(t) isone 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 periodp, equals the Laplace Transform ofone cycleof 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`.

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