# 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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