# Unilateral Laplace Transform's Differentiation Property

I've read in numerous places that the unilateral laplace transform is extermely useful in solving differential equations with initial conditions based on the differentiation property of the unilateral transform:

$$\mathscr{L}{f′(t)}=sF(s)−f(0_−)$$

What i don't understand is why this is possible to do in Laplace but not in the Fourier Transform? Is this related to the dacaying exponential added by the Laplace transform?

I would be grateful for an in-depth answer involving intuition.

• If you look at "transient response" with Laplace xform you will find a lot :) The first real electronics book I read was Transient Response in linear systems. You should realize that these initial terms f(0), f'(0) are the boundary conditions and really are required to analyze a circuit. They can be folded into the Fourier transform but the interpretation/characterization is much harder; and not needed. Be aware that things like Power supplies, motors, and power semiconductors are most stressed during startup; i.e. the worst conditions can occur due to "transients". Mar 30 at 19:40

\begin{align}\mathcal{L}\{f'(t)\}&=\int_{0^-}^{\infty}f'(t)e^{-st}dt\\&=f(t)e^{-st}\Big|_{0^-}^{\infty}+s\underbrace{\int_{0^-}^{\infty}f(t)e^{-st}dt}_{F(s)}\\&=\lim_{t\to\infty}f(t)e^{-st}-f(0^-)+sF(s)\tag{1}\end{align}
The first term in $$(1)$$ is only guaranteed to vanish for $$t\to\infty$$ if $$\text{Re}\{s\}>\alpha$$ for some value $$\alpha$$. So even if you define a unilateral Fourier transform, that term may not vanish in general.
However, if $$\lim_{t\to\infty}f(t)=0$$ holds, we could use the same property with a unilateral Fourier transform. It's just much more common to use the well-established (unilateral) Laplace transform in cases where non-zero initial conditions need to be taken into account.
• Ok thank you very much this is exactly what i was looking for!, and now if you could please complete my analysis: the above method with $0_-$ is meant to deal with transients e.g. a step function - say i've a a basic rc circuit and at time t=0 the switch closes introducing a step function into the system - its derivative is the delta function which also introduces a singularity at t=0, is all this just a method to deal with this singularity of transients? - by modeling the above as if it has initial conditions? thus enabling us to perdict the system's response in full? Mar 27 at 15:14