Initialization

Clear[s, t, ℒ];

Review of Improper Integrals

Here is a standard improper integral from second semester calculus:

(∫_0^∞) (E^(-2 t)) \[DifferentialD]t

Here is a more complicated one. (The previous integral is the s=2 case of this one.)

(∫_0^∞) (E^(-s t)) \[DifferentialD]t

Mathematica is careful about issues of convergence. Basically it is saying that this improper integral converges to 1/s if s > 0 and diverges otherwise. It even considers the case where s may be a complex number.

Some Laplace Transforms

The following improper integral corresponds to the Laplace transform of the exponential function:

(∫_0^∞) (E^(a t)) (E^(-s t)) \[DifferentialD]t

We say that the Laplace transform of the exponential function (e^(a t)) is (1/(s - a)) if s>a.

This improper integral corresponds to the Laplace transform of the sine function:

(∫_0^∞) Sin[
          t] (E^(-s t)) \[DifferentialD]t

The Laplace transform of the sine function is (1/((s^2) + 1)) if s>0.

This improper integral corresponds to the Laplace transform of a product of an exponential and a sine function. Mathematica takes a couple of minutes to worry about issues of convergence.

(∫_0^∞) (E^(2 t)) Sin[
          3 t] (E^(-s t)) \[DifferentialD]t

The result is (3/((s^2) - 4  s + 13))if s>2.

The Laplace Transform Operator

Here is how we can define the Laplace transform operator in Mathematica:

ℒ[
        y_] := (∫_0^∞) 
          y (E^(-s t)) \[DifferentialD]t

Using that definition, we can compute some transforms:

ℒ[Sin[t]]

ℒ[(E^(2  t)) Sin[
          3  t]]

ℒ[(E^(3  t))]                    (* 
      special case of (E^at) *)

ℒ[(t^4)]

ℒ[Sin[2  t]]

ℒ[t Cos[(Sqrt[2]) t]]

Mathematica also has a built-in Laplace transform function which is quicker and gives more readable output:

ℒ[y_] := LaplaceTransform[y, t, s]

ℒ[Sin[t]]

ℒ[(E^(2  t)) Sin[
          3  t]]

ℒ[(E^(3  t))]                    (* 
      special case of (E^at) *)

ℒ[(t^4)]

ℒ[Sin[2  t]]

ℒ[t Cos[(Sqrt[2]) t]]

We will find that the transform of (e^(i ω t)) is useful when we solve second-order equations:

ℒ[(E^(I ω t))]

Partial Fractions Decomposition

We use the following partial fractions decomposition in class:

Apart[(1/((s - 2) (s - 3)))]