  We show how Cauchy's Integral Formula and the ideas of Dunford's Holomorphic
Functional Calculus (for unbounded operators) can be used to compute the Vacuum
Characteristic Function (Quantum Fourier Transform) of quantum random variables
defined as self-adjoint operators on $L^2(\mathbb{R},\mathbb{C})$. We consider
in detail several quantum observables defined in terms of the position and
momentum operators $X$, $P$, respectively, on $L^2(\mathbb{R},\mathbb{C})$.
