1. The Laplace Transform — Definition & Convergence
The Big Idea: The Laplace transform converts a function of time $f(t)$ into a function of a complex variable $s$, by multiplying $f(t)$ by a decaying exponential $e^{-st}$ and integrating over all $t \ge 0$. The payoff: differentiation becomes multiplication — calculus problems become algebra problems.
1.1 The Formal Definition
Let $f(t)$ be defined for $t \ge 0$. Its Laplace transform is
$$F(s) = \mathcal{L}\{f(t)\} = \int_0^{\infty} e^{-st}\, f(t)\, dt$$for those values of $s$ for which the improper integral converges.
This is an improper integral — the upper limit is $\infty$. We interpret it as a limit:
$$F(s) = \lim_{N \to \infty} \int_0^{N} e^{-st}\, f(t)\, dt$$The integral may converge for some values of $s$ and diverge for others.
1.2 The Growth Condition — Exponential Order
A function $f(t)$ is of exponential order $\alpha$ if there exist constants $M > 0$ and $T > 0$ such that
$$|f(t)| \le M\, e^{\alpha t} \quad \text{for all } t > T$$If $f(t)$ is piecewise continuous on $[0, \infty)$ and of exponential order $\alpha$, then $\mathcal{L}\{f(t)\}$ exists for all $s > \alpha$.
Example of failure: The function $f(t) = e^{t^2}$ is not of exponential order — it grows faster than any $e^{\alpha t}$. Its Laplace transform does not exist for any $s$.