# Laplace Transform of Some Basic Functions

1. Exponential Function

By definition of Laplace transform,
Similarly, for f(t) = eâ€“at
1. Unit Step Function or, Heaviside Unit Function
f(t) = u(t) = 1 for t > 0

= 0 for t < 0

and is undefined for t = 0.

(a) Unit step function

(b) Shifted unit step function

Also, the Laplace transform of step function of magnitude K is

Similarly, the Laplace transform of the shifted unit step function u(t â€“ T) is,
{by differentiation theorem}

Another function, called gate function can be obtained from step function as follows.

Gate function

Therefore, g(t and,
1. The Sine Function
f (t) =

F(s) =

1. The Cosine Function
f (t) =

F(s) =

1. The Hyperbolic Sine Function
f (t) =

F(s) =
1. The Hyperbolic Cosine Function
f (t) =

F(s) =

1. The Damped Sinusoidal Function
f (t) =

F(s) =

1. The Damped Cosine Function
f (t) =
F(s) =

1. The Ramp Function
Ramp function

Integrating by parts, let,
u = tn and dv = eâ€“ st dt
then  and

Now, F(s) =

For n =1,

For n =2,
1. Impulse Function or Dirac Delta Function [Î´ (t)]
Generation of impulse function from gate function

It is a function of a real variable t, such that the function is zero everywhere except at the instant t = 0. Physically, it is a very sharp pulse of infinitesimally small width and very large magnitude, the area under the curve being unity.

Consider a gate function as shown in figure.

The function is compressed along the time-axis and stretched along the y-axis, keeping area under the pulse unity. As aâ†’0, the value of  and the resulting function is known as impulse.

It is defined as, Î´ (t) = 0 for t â‰  0
and
Also, Î´ (t) =
The Laplace transform of the impulse function is obtained as,

[Î´ (t)]
= 1