Abstract
We investigate the Lie series representation of the canonical transformations in a complex phase space. It is shown that any canonical mapping in the complex domain can be labelled by two different functions. One of these functions corresponds to an observable in the sense of classical mechanics. The second one has special analytic properties and can be used to form quantities which are important in quantum statistical mechanics. In particular we show that the entropy of ideal quantum gases generates a special canonical transformation and, moreover, the entropy itself can be represented as a Lie function formed by a special characteristic function.