The transfer function of a linear dynamical system is the ratio of the Laplace transform of its output to the Laplace transform of its input. In systems theory, the Laplace transform is called the ``frequency domain'' representation of the system.

Consider a canonical dynamical system \begin{eqnarray*} \dot x(t) &=& A x(t) + B u(t) \\ y (t) &=& C x(t) + D u(t) \end{eqnarray*}with input $u: R \mapsto R^n$ output $y: R \mapsto R^m$ and state $x:R \mapsto R^p$ and $(A,B,C,D)$ are constant matrices of conformable sizes.

The frequency domain representation is $$ y(s) = (D + C(sI - A)^{-1}B)u(s), $$ and thus the transfer function matrix is $D + C(sI - A)^{-1}B$

In the case of single-input-single-output systems ($m=n=1$ , the transfer function is commonly expressed as a rational function of $s$ $$ H(s) = \frac{\prod_{i=0}^Z (s - z_i)}{\prod_{i=0}^P (s - p_i)}. $$ The values $z_i$ are called the zeros of $H(s)$ and the values $p_i$ are called the poles. If any of the poles has positive real part, then the transfer function is termed unstable; if all of the poles have strictly negative real part, it is stable.