# Laplace Transform

Given a function , the Laplace Transform of denoted by (given that the integral exists) is defined by

• is a complex parameter called the frequency

Why is s called the frequency?

In , the term relates to angular frequency in radians per second.

The teacher gives these two examples:

Laplace Transforms of common functions:

### Properties of Laplace Transform

Theorem 1

For any function , the one-sided Laplace transform will always converge for some that is sufficiently large.

Theorem 2: Laplace Transform is Linear

Suppose that and have Laplace transforms and . Then for all

the the ROC is the intersection of the ROCs for and .

Theorem 3: Time-Scaling

If then for

Theorem 4: Exponential Modulation

Theorem 5: Time-Shifting

If and then

Theorem 6: Multiplication by t

If then

Theorem 7: Laplace Transform of a Derivative/Integral

Let be such that there is a real value α such that the integral

converge and such that there exists a function such that for

and there exists a real value such that

converges. In this case or in other-words

• We can use this theorem to solve linear ODEs!