ChapterZero

Laplace Transforms

Filed under: General — Alex @ 8:14 pm 11/4/2003

I just got out of my circuits and systems class, in which I paying no attention. Instead, I was fiddling around trying to find the inverse laplace transforms of some functions. It all started when a guy asked me what the ILT of $\frac{s}{s+1}$ is; it took me a short while, but I soon worked it out by using division to rewrite it as $1 - \frac{1}{s+1}$ and taking the term by term inverse to get $\delta(t) - e^{-t}$ . I originally tried using the derivative rule, that $L\{df/dt\} = sF(s) - f(0^-)$ , but I started confusing myself. Then I was intrigued, and tried generalizing the division process to help me find the ILT of a function in the form of a polynomial in over s+1 . This led me to consider the inverses of s^n , which turned out to be the -th derivatives of $\delta(t)$ , from the derivative rule.

Near the end of class, I started trying more exotic functions like cos(s) and tan(s) , e^s , which I suspect don’t have ILTs. I had the thought of expanding them into Taylor Series and using the term by term inverses, but is that sensible? I mean, does the linearity of the transform extend to cover infinite sums? I suspected not, just because that would make life too simple And after checking the Internet just now for the conditions under which an IFT exists, I think I was right for suspecting so; certainly, e^s doesn’t have an IFT, because it doesn’t satisfy the two necessary conditions: