We show that the technique of integration within an ordered product of operators can be extended to Hilbert transform. In so doing we derive normally ordered expansion of Coulomb potential-type operators directly by using the mathematical Hilbert transform formula.
Two types of canonical transformations in two-mode classical phase space are mapped into the quantum mechanical Hilbert space to produce some new normally ordered unitary operators. These operators are evaluated in the coordinate (momentum) representations using the "integration within ordered product" technique, and the mapping is manifestly apparent in the derivation. New generalized coherent states are constructed in terms of these operators, and the uncertainty relations for these states are analysed.