Converting continuous-time transfer functions such as and to the digital domain is analogous to converting an analog electrical filter to a corresponding digital filter--a problem which has been well studied . For this task, the bilinear transform (§7.3.2) is a good choice. In addition to preserving order and being free of aliasing, the bilinear transform preserves the positive-real property of passive impedances (§C.11.2).
Digitizing via the bilinear transform (§7.3.2) transform gives
which is a second-order digital filter having gain less than one at all frequencies--i.e., it is a Schur filter that becomes an allpass as the damping approaches zero. The choice of bilinear-transform constant maps the peak-frequency without error (see Problem 4).