@Chen Siyi

November 2, 2020

Note6 Laplace Transform

The (Unilateral) Laplace Transform

Definition of (Unilateral) Laplace Transform

Let $f : [0, ∞) → \mathbb{R}$ be a continuous function such that

Then the function $F : (β, ∞) → \mathbb{R}$,

is called the (Unilateral) Laplace transform of $f$ .

A Reminder

Why should the "sup" condition be satisfied for the (Unilateral) Laplace transform?

For any $f$, will there always exist a region of $p$ where $F(p)$ is defined?

And the ROC(radius of convergence) is the region for $p$, where $F(p)$ converges. So the (Unilateral) Laplace Transform $(\mathscr{L} f)(p)$ is defined for such $p$.

A Tricky Question

Can the ROC of the (Unilateral) Laplace Transform of a function $f$ be in the form $(-∞, \beta)$?

Definition of (Bilateral) Laplace Transform

The bilateral Laplace transform is defined as

Of course, we have

The following questions may help you understand better the relationship between these two types of Laplace Transform.

A Tricky Question

What should the value of $p$ satisfy for the (Bilateral) Laplace Transform of $f$ to exist?

A Tricky Question

If for a funstion $f$, the Unilateral Laplace Transform of $f$ is defined in some region of $p$, will there always be a Bilateral Laplace Transform for certain region of $p$?

A Tricky Question

Can the ROC of the (Bilateral) Laplace Transform of a function $f$ be in the form $(-∞, \beta)$, $(\alpha, \beta)$?

*Our main focus here (VV286) is the (Unilateral) Laplace Transform.

Question

Find the Laplace transform of a function of the form $t^a$ with $a > −1$.

Hint:

• We focus on the Unilateral Laplace transform witout stating out.
• Considering the gamma function, what's its definition?

Answer

Properties of (Unilateral) Laplace Transform

• $\mathscr{L}(f * g)=(\mathscr{L} f) \cdot(\mathscr{L} g)$
• $\mathscr{L}(\int_0^t f(\tau) d\tau) = \frac{1}{p}\mathscr{L}(f(t))$

The Inverse of the Laplace Transform

The laplace transform can be extended to the complex plane, by using the analytic continuation where we set $F(z)$ by replacing $p$ with $z$ in $F(p)$ for $Rez > \beta$. (Why?)

For $z = p + iq$, where $p > \beta$ we have

1. $F(z)$ exists.
2. $F(z)$ is holomorphic. Since the integral converges absolutely, and the function $g(z) = e^{zt}$ is complex differentiable for any $t ∈ [0, ∞)$. (Recall what is $F$).

The Bromwich Integral

Definition

Let $Ω ⊂ \mathbb{C}$ be an open set, $β ∈ \mathbb{R}$ and $F : Ω → \mathbb{C}$ analytic for all $z ∈ \mathbb{C}$ with $Re z ≥ β$. Then the Bromwich integral of $F$ is defined as

where $\mathcal{C} = {z ∈ C: Rez = β}$ is the Bromwich contour, oriented positively if the contour is closed on the left (i.e., the line is traversed in the direction of positive imaginary part.)

Often, the integral is written

An example for the Bromwich integral of a function $F$ is given below.

A Tricky Question

Does it matter which $\beta$ you choose?

--Yes and no.

A Reminder

Where $F(p)$ is a Unilateral Laplace Transform, and the Bromwich contour must be located within the ROC. So it will always be holomorphic in the right part of the Bromwich integral.

Such $f(t)$ is causal because there are no poles to the right of the Bromwich contour.

Evaluation

Summary

1. The region of $t$, i.e. whether $t < 0$ or $t > 0$, decides which semi-circle you would like to use for integral.

Why?

2. The positions of poles, decide which theorem to use for integral.

Which theorems? How do you decide?

Concrete Analysis

• $t > 0$

  $$\int_{\Gamma_{\beta, R}} e^{p t} F(p) d p = i e^{\beta t} \int_{C_{R}} e^{i t p} F(\beta+i p) d p$$

Jordan's lema shows the integral would vanish as $R → ∞$.

Use the Residue Theorem for this specific case.

• $t < 0$

  $$\int_{\Gamma_{\beta, R}^{(2)}} e^{p t} F(p) d p = i e^{\beta t} \int_{C_{R}} e^{i|t| p} F(\beta-i p) d p$$

  Jordan's lema shows the integral would vanish as $R → ∞$.

  Use Cauthy's Integral Theorem for this specific case.

A Tricky Question

"Whether the integral along the two semi-circles vanishes", is it related to the poles of F?

The Mellin Inversion Formula

If $f$ is continuous on $[0, ∞)$, continuously differentiable on $(0, ∞)$ and satisfies $\sup _{t \in[0, \infty)} e^{-\beta t}|f(t)|<\infty$ for some $\beta > 0$, then

which is called the Mellin inversion formula for the Laplace transform.

Question

Let $F(p) = 1/p$ be the Laplace transform of a time signal $f(t)$ with the half-plane Re(s) > 0 as its ROC. Find $f(t)$ using the Mellin inversion formula.

Hint:

• We focus on the Unilateral Laplace transform witout stating out.
• The Bromwich contour must be located within the ROC
• *What happens to the point $t = 0$ ?

Answer

Solving Differential Equations with Laplace Transform

Overall Idea

1. Apply the Laplace transform to both sides of the ODE/IVP.

2. Solve for the Laplace transform $Y$.

3. Find inverse Laplace transform, which is the solution. You can try (in mixture of) several ways:

   1. Decomposite $Y(p)$ into partial fractions.
   2. Use the table of pairs, and also properties.
   3. Use the mellin inversion formula.

Question

Solve the IVP:

$$y^{\prime \prime}-10 y^{\prime}+9 y=5 t, \quad y(0)=-1 \quad y^{\prime}(0)=2$$

Answer

Using Green's Function and Convolution

• A Convolution Product for the Laplace Transform:

• A linear, second order, inhomogeneous ODE with constant coefficients:

1. Apply the Laplace transform to both sides of the ODE/IVP.

2. Solve for the Laplace transform $Y$.

   $$Y=(\mathscr{L} f)(p) \cdot \frac{1}{a p^{2}+b p+c}+\frac{a y_{0} p+b y_{0}+a y_{1}}{a p^{2}+b p+c}$$

3. Find the Green's Function g(x) such that $(\mathscr{L} g)(p)=\frac{1}{a p^{2}+b p+c}$

4. Use transform table and apply convolution to find inverse Laplace transform, which is the solution.

Question

Solve the IVP:

$$y^{\prime \prime}+y=\left\{\begin{array}{l} \cos t, 0 \leqslant t \leqslant \pi / 2 \\ 0, \pi / 2 \leqslant t<\infty \end{array}=\cos t \cdot H\left(\frac{\pi}{2}-t\right), y(0)=3, y^{\prime}(0)=-1\right.$$

Answer

Impulses and the Delta Function