# 4. Laplace Transforms of the Unit Step Function

We saw some of the following properties in the Table of Laplace Transforms.

Recall `u(t)` is the unit-step function.

1. `ccL{u(t)}=1/s`

2. `ccL{u(t-a)}=e^(-as)/s`

3. Time Displacement Theorem:

If `F(s)=ccL{f(t)}` then `ccL{u(t-a)*g(t-a)}=e^(-as)G(s)`

[You can see what the left hand side of this expression means in the section Products Involving Unit Step Functions.]

### Examples

Sketch the following functions and obtain their Laplace transforms:

(a) `f(t)={ {: (0,t < a), (A, a < t < b), (0, t > b) :}`

Assume the constants *a*, *b*, and *A* are positive, with *a* < *b*.

(b) `f(t)={ {: (0,t < a), (e^(t-a), a < t < b), (0, t > b) :}`

Assume the constants *a* and *b *are positive, with *a* < *b*.

(c) `f(t)={ {: (0,t < 0), (sin\ t, 0 < t < pi), (0, t > pi) :}`

Didn't find what you are looking for on this page? Try **search**:

### Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Go to: Online algebra solver

### Ready for a break?

Play a math game.

(Well, not really a math game, but each game was made using math...)

### The IntMath Newsletter

Sign up for the free **IntMath Newsletter**. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

### Share IntMath!

### Calculus Lessons on DVD

Easy to understand calculus lessons on DVD. See samples before you commit.

More info: Calculus videos