Section 4 - Laplace Transformations

Section 4.1 - Definition

This section is from Paul's Online Math Notes.

Definition. The Laplace transform of a function is given by the following:

\[ \mathcal{L} \{f(t)\}(s) = F(s) = \int_0^{\infty} e^{-st}f(t) dt \]

Section 4.2 - Properties

The Laplace Transformation is a linear transformation over functions in \(\mathbb{R}[t]\). That is, given \(a, b \in \mathbb{R}, f(t), g(t) \in \mathbb{R}[t]\), we know that

\[ \mathcal{L} \{a f(t)\ + b g(t) \}(s) = a F(s) + b G(s) \]

Section 4.3 - Inverse Laplace Transformation

Given \(F(s)\), we define the Inverse Laplace Transformation as the following;

\[ f(t) = \mathcal{L}^{-1} \{F(s)\} \]

Section 4.4 - Step Function

The step/Heaviside function \(u_c(t)\) is defined as 0 if \(t < c\), and 1 if \(t > c\).

Alternatively, \(u(t - c) = H(t - c)\) is 0 if \(t < c\), and 1 if \(t > c\).

Applying this to the Laplace transform,

\[ \begin{align} \mathcal{L} \{ u_c(t) f(t-c) \} &= \int_0^{\infty} e^{-st}u_c(t)f(t) dt \\ &= \int_c^{\infty} e^{-st}f(t) dt \end{align} \]

If we let \(u = t - c\),

\[ \begin{align} \mathcal{L} \{ u_c(t) f(t-c) \} &= \int_0^{\infty} e^{-s(u+c)}f(u) du \\ &= \int_0^{\infty} e^{-su}e^{-cs}f(u) du \\ &= e^{-cs} \int_0^{\infty} e^{-su}f(u) du \\ &= e^{-cs} F(s) \end{align} \]

Section 4.5 - Laplace Transformation applied to IVPs

Theorem. Given a function \(f(t)\) with \(C^n\) continuity, then

\[ \mathcal{L} \{ f^{(n)} (t) \} = s^n F(s) - s^{n-1} f(0) - s^{n-2} f'(0) - \ldots - s f^{(n-2)} (0) - f^{(n-1)} (0) \]

For \(n=1, 2\) we see that

\[ \begin{align} \mathcal{L} \{ y' \} &= sY(s) - y(0) \\ \mathcal{L} \{ y'' \} &= s^2 Y(s) - s y(0) - y'(0) \end{align} \]

We can take the Laplace transformation of an IVP, solve for \(Y(s)\), then take the inverse to find the solution.

Section 4.6 - Non-constant Coefficient IVPs

If \(f(t)\) is piecewise continuous on \([0, \infty)\), then \(\lim_{s \rightarrow \infty} F(s) = 0\).

Definition. A function \(f(t)\) is said to be of exponential order \(\alpha\) if there exists positive constants \(T, M\) such that for all \(t \geq T\), \(|f(t)| \leq Me^{\alpha t}\).

To check this, simply compute \(\lim_{t \rightarrow \infty} \frac{|f(t)|}{e^{\alpha t}}\). If this is finite for some \(\alpha\), then the function is of exponential order \(\alpha\).

Section 4.7 - IVPs with Step Functions

Recall that \(\mathcal{L} \{u_c(t)f(t-c)\} = e^{-cs}F(s)\). Then, we can solve IVPs involving step functions.

Section 4.8 - Dirac Delta Function

The Dirac Delta function has several properties. First, \(\delta(t - a) = 0\) when \(t \neq a\). Notably, though,

\[\int_{\mathbb{R}} f(t) \delta(t - a) dt = f(a)\]

Note that this is not an actual function, buy instead a generalized function or distribution, as several functions can express this property using infinite limits.

Then, we can see that \(\mathcal{L} \{\delta(t-a)\} = \int_0^\infty e^{-st} \delta(t-a) dt\) by definition. Then, applying the properties of the Delta function, \(\mathcal{L} \{\delta(t-a)\} = e^{-as}\), given \(a > 0\).

Section 4.9 - Convolution Integrals

Consider two functions \(F(s)\) and \(G(s)\) such that \(F(s) G(s) = H(s)\), of which we want to find an inverse Laplace transform.

We define a convolution integral \((f*g)(t)\) as

\[(f*g)(t) = \int_0^t f(t - \tau)(g - \tau) d\tau\]

A unique property of this integral is that \((f*g) = (g*f)\).

With this, we see that \(\mathcal{L} \{f * g\} = F(s)G(s)\), or that \(\mathcal{L}^{-1} \{F(s)G(s)\} = (f * g)(t)\).