Laplace Transform Properties

As we saw from the Fourier Transform, there are a number of properties that can simplify taking Laplace Transforms. I'll cover a few properties here and you can read about the rest in the textbook.

Real Time Shifting

Derive this:

Plugging in the time-shifted version of the function into the Laplace Transform definition, we get:

Letting τ = t - t₀, we get:

Example 1  Find the Laplace Transform of x(t) = sin[b(t - 2)]u(t - 2)

Differentiation

Recall the equation for the voltage of an inductor:

If we take the Laplace Transform of both sides of this equation, we get:

which is consistent with the fact that an inductor has impedance sL.

Proof of the Differentiation Property:

1) First write x(t) using the Inverse Laplace Transform formula:

2) Then take the derivative of both sides of the equation with respect to t (this brings down a factor of s in the second term due to the exponential):

3) This shows that x'(t) is the Inverse Laplace Transform of s X(s):

The Differentiation Property is useful for solving differential equations.

Integration

Recall the equation for the voltage of a capacitor turned on at time 0:

which is consistent with the fact that a capacitor has impedance

.

Additional Properties

Multiplication by t

Derive this:

Take the derivative of both sides of this equation with respect to s:

This is the expression for the Laplace Transform of -t x(t). Therefore,

Initial Value

(Given without proof)

Final Value

(Given without proof)

Independent-Variable Transformation (for Unilateral Laplace Transform)

Derive this:

Plugging in the definition, we find the Laplace Transform of x(at -b):

Let u = at - b and du = adt, we get: