transfer function - What is the relationship between poles and system stability?

I see two notions that describe the relationship between poles and system stability. But they are not the same from my understanding

1. The system is BIBO stable if and only if all the poles are in the left half of the complex plane


2. A LTI system with a rational system function H(z) is stable if and only if all of the poles of H(z) lie inside the unit circle.

Why these two notions are different? Is that in different conditions?

Answer

The two are both true, but they are for different cases. Case 1 is true for continuous-time systems, and the transform is the Laplace transform and the variable is the derivative operator, $s$. Case 2 is true for discrete-time systems, and the transform is the $z$-transform and the variable is the delay operator, $z$.