Region of Convergence and Existence

The region in the complex plane (set of values $s$) for which $\mathcal{L}$ or $Z$ converges or exists.

For example: $h(t)=e^{-at}u(t)$

Then, we have:

$$\begin{array}{rl}H(s)& =\mathcal{L}[h(t)]={\int }_{-\infty }^{\infty }h(t)e^{-st}dt\\ & ={\int }_{-\infty }^{\infty }{e}^{-at}{e}^{-st}u(t)dt\\ & ={\int }_0^{\infty }{e}^{-(s+a)t}dt\\ & =-\frac{1}{s+a}{e}^{-(s+a)t}{|}_0^{\infty }\\ & =-\frac{1}{s+a}\underset{t\to \infty }{\lim }{e}^{-(s+a)t}-1\end{array}$$

For the limit in the final expression, we have:

$$\underset{t\to \infty }{\lim }{e}^{-(s+a)t}=\{\begin{array}{llc}0& \text{if}& \text{Re}[-(s+a)]<0\\ \infty & \text{if}& \text{Re}[-(s+a)]>0\end{array}$$

This can be simplified to:

$$\begin{array}{rl}\text{Re}[-(s+a)]& <0\\ \text{Re}(-a)+\text{Re}(-s)& <0\\ -a+\text{Re}(-s)& <0\\ \text{Re}(-s)& <a\\ -\text{Re}(s)& <a\\ \text{Re}(s)& >-a\end{array}$$

This is the region of convergence, such that

$$H(s)=\frac{1}{s+a}\text{for}\underset{\text{ROC}}{\underbrace{\text{Re}(s)>-a}}$$