A question about the idea of Laplace transform

Question

In Laplace integral transform equation one multiplies the function $f(t)$ by $e^{-st}$. I read in many tutorials that $e^{-st}$ decays much faster than any other function so the integral diverges.

But what I don't understand what makes one to think to multiply a function with $e^{-st}$ to transform it in $s$ domain? What is the motivation behind it? Why would you suddenly come up with an idea as such: "Oh there is a function $f(t)$ in time domain what can I do to transform it to complex freq. ($s$) domain? Hmm let me multiply it with $e^{-st}$ and integrate it from zero to infinity" What would have motivated this idea of multiplying and integrating for transformation?

Answer 1

If we're already used to the idea of relating an infinite sequence $a_n$ to the infinite series $\sum_{n=0}^\infty a_n x^n,$ then it might seem natural to relate a function $a(t)$ of the continuous variable $t\in[0,\infty)$ to the infinite integral $\int_{t=0}^\infty a(t)x^tdt.$ However, since we are used to writing exponential functions to the base $e$ with a coefficient in the exponent, and in order to make the integrations less awkward, it's convenient to substitute $x=e^s$ and write $\int_{t=0}^\infty a(t)e^{st}dt.$ This is the familiar Laplace transform of $a(t)$ except that the region of convergence is on the negative $s$-axis. Since we are prejudiced in favor of positive numbers, we fix this problem by changing the sign of $s,$ so the Laplace transform of $a(t)$ is defined as $A(s)=\int_{t=0}^\infty a(t)e^{-st}dt.$

Answer 2

Denote by ${\cal R}$ (for response) the space of functions $f:\>{\mathbb R}_{\geq0}\to{\mathbb C}$ that are sufficiently smooth and do not increase faster than some exponential function when $t\to\infty$. Initial value problems of (inhomogeneous) linear ODEs with constant coefficients typically have their solutions in ${\cal R}$, hence this is an important space in applications.

Consider the innocuously looking functions $$E_s:\> t\mapsto e^{-st}\qquad (s\in{\mathbb C})$$ as probes which we use in order to make measurements on individual known or as yet unknown functions $f\in{\cal R}$. The measurement takes the following form: We compute the integral $$F(s):=\int_0^\infty f(t)\>E_s(t)\>dt=\int_0^\infty f(t)\>e^{-st}\>dt\ .$$ Since $f$ by assumption increases at most exponentially the value $F(s)$ is defined for all complex $s$ with ${\rm Re}\, s>s_0$. This implies that for any $f\in{\cal R}$ its Laplace transform $s\mapsto F(s)$ is defined in a full half plane, hence carries enough information to recover $f$ from $F$ (Lerch's theorem).

Why should we do this? The central point is that the transform $L:\>f\mapsto F$ has phantastic formal, resp., analytical properties. It transforms a problem involving an unknown function $f$ into a simpler problem involving $F$. In particular differentiation in the $t$-world appears as multiplication with $s$ in the $s$-world. Etcetera.