**Introduction**

**Unit Step Function**

The unit step function at time t, *u*(*t*), is defined as

That is, u or H is a function of time t,

- When time is negative, u have a value zero,
- When time is positive, u have a value one.

At t = 0,

- In some textbooks u(0) = 1, and somewhere it is 0.5.
- We are considering the second case, but it won't make much difference.

The thing is, u(t) is discontinuous at 0, therefore not differentiable at 0. The unit step function is also known as a Heaviside unit step function.

**Shifted Unit Step Function**

A shifted unit step function has value 0 up to the time *t *= *a* and has value 1 afterward.

**Laplace Transformation**

A function is considered piecewise continuous if it has a finite number of breaks and does not blow up anywhere. If the function f(t) is a piecewise continuous function, then the Laplace transform is used to define it. A function's Laplace transform is denoted by Lf(t) or F. (s). The Laplace transform helps solve differential equations by converting them into algebraic problems.

Laplace transform of a function f(t) is given by the equation:

**Laplace transform of a unit step function**

**Step 1:** Formula of Laplace transform for f(t).

**Step 2:** Unit Step function u(t):

**Step 3:** Now, as the limits in Laplace transform goes from 0 -> infinity, u(t) function = 1 in the interval 0 -> infinity. Hence Laplace transform equation for u(t):

Solving the above integral equation gives,

Hence, Laplace transform of a unit step function is