Confusion in basics of Laplace Transform
The unilateral Laplace transform is used for analyzing causal linear time-invariant systems, which have an impulse response $h(t)$ that is zero for $t&lt;0$. The unilateral Laplace transform can be used to solve initial value problems, due to the correspondence

$$x'(t)\Longleftrightarrow sX(s)-x(0)$$

where $x(0)$ is a given initial value for the function $x(t)$. Note that for the bilateral Laplace transform the equivalent correspondence is simply $x'(t)\Longleftrightarrow sX(s)$.

Concerning the signal $x(t)=e^{-at}u(t)$, note that its Laplace transform is

$$X(s)=\int_0^{\infty}e^{-(a+s)t}dt$$

This integral only converges if the exponential decays, which results in the condition $\text{Re}(a+s)&gt;0$. This condition defines the region of convergence (ROC) of the Laplace transform. $X(s)$ only exists for values of the complex variable $s$ satisfying $\text{Re}(s)&gt;-\text{Re}(a)$.