# 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. ℒ`{u(t)}=1/s`

2. ℒ`{u(t-a)}=e^(-as)/s`

3. Time Displacement Theorem:

If `F(s)=` ℒ`{f(t)}` then ℒ`{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) :}`

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