Laplace Transform of Standard Functions

Laplace Transform of Standard Functions:

The Laplace Transform of Standard Functions is given by

Step Function:

The unit step function is,

Laplace Transform of Standard Functions

If there is step of amplitude A i.e. f(t) = A u(t) then its Laplace is given by A/s.

Laplace Transform of Standard Functions

If the unit step is delayed by T instants then,

From the shifting theorem of Laplace transform we can write,

Laplace Transform of Standard Functions

Ramp Function:

The unit ramp function is defined as,

Laplace Transform of Standard Functions

Integrating by parts,

Laplace Transform of Standard Functions

While the Laplace transform of ramp having slope A i.e.

Laplace Transform of Standard Functions

If the unit ramp is shifted by T instants then,

Laplace Transform of Standard Functions

From the shifting theorem of Laplace transform,

Impulse Function:

The unit impulse function is δ(t) and defined as,

Laplace Transform of Standard Functions

We know the relation between unit step and unit impulse.

Taking Laplace transform of both sides,

Laplace Transform of Standard Functions

If there is a delayed impulse function δ(t – T) then using shifting property we can write,

Laplace Transform of Standard Functions

Let us summarise these results in the tabular form so that while finding the Laplace transform of the various time functions this table of standard Laplace transforms can be referred directly.

Laplace Transform of Standard Functions

Thus the Laplace transforms of standard time functions are very useful to obtain the Laplace transform of any waveform by resolving it into standard function.

Updated: October 11, 2019 — 10:36 pm