How to derive stability of analogue and digital systems I'm trying to find out why digital systems are stable iff poles oh their transfer function are inside the unit circle (or are there any others conditions?).

I understand analogous systems are stable iff all poles of their transfer function are negative 'cause of inverse Laplace transform, where $\mathcal{L}^{-1} \left \{ \frac {1}{s+a} \right \} = e^{-at} \cdot \mathcal{1} (t) $. (by the $\mathcal{1} (t)$ i mean heaviside function). Is that right?

But how to derive conditions for stability of discrete system?

What others ways for derivating stability of discrete and continuous systems exist?
In the discrete-time case you can use the $\mathcal{Z}$-transform:

$$\mathcal{Z}\{x(n)\}=X(z)=\sum_{n=0}^{\infty}x(n)z^{-n}\tag{1}$$

Equation (1) is the unilateral $\mathcal{Z}$-transform which is useful for causal signals which are zero for $n&lt;0$. Now consider the signal $x(n)=a^n$, $n\ge 0$:

$$X(z)=\sum_{n=0}^{\infty}a^nz^{-n}$$

This series converges for $|a|&lt;1$ and in this case $X(z)$ is given by:

$$X(z)=\frac{1}{1-az^{-1}}=\frac{z}{z-a}\tag{2}$$

From (2) you can see that $a$ is the pole of $X(z)$, and for $x(n)$ to be stable (if seen as a system's impulse response), and for $X(z)$ to exist, $|a|&lt;1$ must hold, i.e. the pole must be inside the unit circle of the complex $z$-plane. This example is analogous to the continuous-time example in your question.

Also take a look at the concept of BIBO-stability (bounded-input-bounded-output stability), which leads to the same conclusion: for continuous time signals/systems the poles must be in the strict left half of the $s$-plane, and for discrete-time signals/systems the poles must be strictly inside the unit circle of the $z$-plane. The condition on the time-domain signals is that they be absolutely integrable in the continuous case, and absolutely summable in the discrete-time case:

$$\int_{-\infty}^{\infty}|x(t)|dt &lt;\infty\\
\sum_{n=-\infty}^{\infty}|x(n)| &lt;\infty$$