The fundamentals of Fan's method of integration within ordered products are
investigated and it is shown that it is a reliable calculation method in q
uantum optics. Differentiation within ordered products is included into the
considerations. The integrations are, in applications to single boson mode
s, mostly one-dimensional or two-dimensional integrations over phase-space
variables. We examine some examples of integration within ordered products
and compare them with calculations of results by other methods such as Lie
group methods. A new representation of the main class of quasiprobabilities
including the Wigner and the Husimi-Kano quasiprobability is derived by th
e method. Different representations of squeezing operators are related by t
he method and discussed. As a further new result, the position and momentum
representation of the general unitary squeezing operators are derived and
specialized to cases with squeezing in the direction of the coordinate axes
which were earlier dealt with by the method under consideration.