The Laplace transform was first introduced by Pierre-Simon Laplace in his work on probability theory. The transform turns out to have many applications in science and engineering, especially in solving linear differential equations.

The Laplace Transform

Let $f(t)$ be defined for $0\leq t<\infty$. The Laplace transform of $f(t)$, which is denoted by $F(s)$, or $\mathcal{L}\left\{f(t)\right\}$ is given by

where the improper integral is understood as

Remarks:

Two conditions that $f$ should be satisfied to have a Laplace transform (for sufficiently large $s$) are

1. The function $f(t)$ is of exponential order, i.e., there exist constants $M$ and $c$ such that for all $0\le t<\infty$,

2. The function $f(t)$ is piecewise continuous, i.e., $f(t)$ has at most a finite number of discontinuities on any interval $0\le t\le A$, and both the limit from the right and the limit from the left of $f$ exist at every point of discontinuity.

The Laplace transform doesn’t care how the function $f(t)$ behaves when $t<0$, this means that different functions can have the same Laplace transform as long as they agree on $t\ge 0$.

Properties

In the following, we will assume $\mathcal{L}\left\{f(t)\right\}=F(s)$, $\mathcal{L}\left\{g(t)\right\}=G(s)$, or denotes as $f(t)\rightsquigarrow F(s)$, $g(t)\rightsquigarrow G(s)$. We have the following important properties.

1. Linearity

2. Frequency domain derivatives

3. Time domain derivatives

4. Frequency shifting

5. Time shifting

where $u(t)$ is the Heaviside step function (unit step function), and $a>0$.

6. Time scaling

where $a>0$.

6. Convolution

The convolution between two functions is defined as

Then, the Laplace transform of the convolution is the ordinary product of the Laplace transforms,

Laplace Transform of Common Functions

The following lists the Laplace transform of some common functions:

1. Dirac Delta function (unit impulse)

2. Heaviside step function (unit step)

3. Polynomial

For integers$n$, the following two formulas hold

where

is the Euler’s Gamma function, with $\Gamma(1/2)=\pi$.

4. Exponential decay

5. Sine and Cosine

6. Hyperbolic Sine and Cosine

Inverse Laplace Transform

The inversion of the Laplace Transform is given by the Mellin inversion formula. In particular, if $f$ is continuous on $\left[0,\infty\right)$, continuously differentiable on $\left(0,\infty\right)$, and satisfies the condition for the Laplace transform to exist, then

for all $s\in\left[0,\infty\right)$, where

is called the Bromwich integral of $F$, and the (positively oriented) contour is defined as

where $\beta\in\R$ is any number such that $F$ is analytic for all $z\in\mathbb{C}$ with $\mathrm{Re}z\ge\beta$.

To calculate the Bromwich integral, we should consider the case when $t\ge0$ and $t<0$ separately.

For $t\ge 0$, the integration contour closed on the left.

Let $\Gamma$ be the straight line and $\mathcal{C}_{\beta,R}$ be the semicircle (orientation indicated by the arrows). Then, by the residue theorem

Letting $R\to \infty$, the first integral be the Bromwich integral we want. For the second integral, by applying the substitution rule,

where $\mathcal{C}_R$ is a semi-circle of radius $R$ in the upper half-plane. In most cases, we can use Jordan’s Lemma to show that the right-hand side goes to 0 as $R\to \infty$.

Then, the Bromwich integral of $F$ is just the sum of all the residues of $F$,

For $t<0$, the integration contour is closed on the left,

Let $\Gamma$ be the straight line and $\mathcal{C}_{\beta,R}^{(2)}$ be the semi-circle (orientation indicated by the arrows). Using a similar argument, we see that

Since the integral over the whole closed contour is zero, and the orientation of the straight line is in the opposite direction, the Bromwich integral is

which in most cases vanishes according to Jordan’s Lemma.

Reference

[1]. VV286 Lecture Slides, Horst Hohberger, UM-SJTU JI.