Definitions and Examples

DEFINITION 10.2.1 (Piece-wise Continuous Function)  

1. A function $f(t)$ is said to be a piece-wise continuous function on a closed interval $[a, b] \subset {\mathbb{R}}$ , if there exists finite number of points $a = t_0 < t_1 < t_2 < \cdots < t_N = b$ such that $\; f(t)$ is continuous in each of the intervals $(t_{i-1}, \; t_i)$ for $1 \leq i \leq N$ and has finite limits as $t$ approaches the end points, see the Figure 10.1.
2. A function $f(t)$ is said to be a piece-wise continuous function for $t \geq 0$ , if $f(t)$ is a piece-wise continuous function on every closed interval $[a, b] \subset [0, \infty).$ For example, see Figure 10.1.

Figure 10.1: Piecewise Continuous Function

DEFINITION 10.2.2 (Laplace Transform)   Let $f: [0, \infty) \longrightarrow {\mathbb{R}}$ and $s \in {\mathbb {R}}.$ Then $F(s),$ for $s \in {\mathbb{R}}$ is called the LAPLACE TRANSFORM of $f(t),$ and is defined by

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

whenever the integral exists.
(Recall that $\int\limits_{0}^\infty g(t) dt$ exists if $\lim\limits_{b \longrightarrow \infty} \int\limits_0^b g(t) d(t)$ exists and we define $\int\limits_{0}^\infty g(t) dt = \lim\limits_{b \longrightarrow \infty} \int\limits_0^b g(t) d(t)$ .)

Remark 10.2.3  

1. Let $f(t)$ be an EXPONENTIALLY BOUNDED function, i.e.,
   $$\vert f(t) \vert \leq M e^{{\alpha}t}\;\; {\mbox{ for all }} \;\... ... real numbers }} \; {\alpha}\; {\mbox{ and }} \; M \; {\mbox{ with }} \; M > 0.$$
   Then the Laplace transform of $f$ exists.
2. Suppose $F(s)$ exists for some function $f$ . Then by definition, $\lim\limits_{b {\longrightarrow}\infty} \int_0^{\mathbf b}f(t) e^{-s t} dt$ exists. Now, one can use the theory of improper integrals to conclude that
   $$\lim\limits_{s {\longrightarrow}\infty} F(s) = 0.$$
   Hence, a function $F(s)$ satisfying
   $$\lim\limits_{s {\longrightarrow}\infty} F(s) \; {\mbox{ does not exist or}} \; \lim\limits_{s {\longrightarrow}\infty} F(s) \ne 0,$$
   cannot be a Laplace transform of a function $f$ .

DEFINITION 10.2.4 (Inverse Laplace Transform)   Let ${\mathcal L}(f(t)) = F(s)$ . That is, $F(s)$ is the Laplace transform of the function $f(t).$ Then $f(t)$ is called the inverse Laplace transform of $F(s)$ . In that case, we write $f(t) = {\mathcal L}^{-1}(F(s)).$