We discuss how the Hamiltonian changes in quantum canonical transformations. To the operator $\stackrel{^}{\mathcal{H}}(\stackrel{^}{p},\stackrel{^}{q})$ one can associate (in a given ordering rule) a
c-number function
H(p,q). It is this function that appears in the action of the phase-space path integral. A quantum canonical transformation $\stackrel{^}{\mathcal{H}}\ensuremath{\rightarrow}{\stackrel{^}{\mathcal{H}}}^{\ensuremath{'}}$ can now be expressed as an integral transformation ${\mathcal{H}}^{\ensuremath{'}}(\overline{p},\overline{q})=\ensuremath{\int}\mathrm{dp}\mathrm{dq}\mathcal{T}(\overline{p},\overline{q};p,q)\mathcal{H}(p,q)$. The kernel
T is constructed explicitly for point transformations and for the $p=\ensuremath{-}\overline{q}$,
q=p reflection by studying changes of variables in the path integral. The ordering dependence of
T is displayed. The invariance of commutation rules is also discussed.
Integral transformations are useful mathematical tool to work out signals and wave-packets in electronic devices. They may be used in software protocols. Necessary knowledge may come from quantum field theory, in particular from quantum chromodynamics, in which the optic theorem and the renormalization group equation can be solved by a unique contour integral written in two different "dual" ways related between themselves by a complex map in the complex plane of Mellin variable. The inverse integral transformation should be modified to be applied for these contour integral solutions. These modified inverse transformations may be used in security protocols for quantum computers. Here we do a brief review of the basic integral transforms and propose their modification for the extended domains.