  The central impediment to reducing multidimensional integrals of transition
amplitudes to analytic form, or at least to a fewer number of integral
dimensions, is the presence of magnitudes of coordinate vector differences
(square roots of polynomials) $|{\bf x}_{1}-{\bf
x}_{2}|^{2}=\sqrt{x_{1}^{2}-2x_{1}x_{2}\cos\theta+x_{2}^{2}}$ in disjoint
products of functions. Fourier transforms circumvent this by introducing a
three-dimensional momentum integral for each of those products, followed in
many cases by another set of integral transforms to move all of the resulting
denominators into a single quadratic form in one denominator whose square my be
completed. Gaussian transforms introduce a one-dimensional integral for each
such product while squaring the square roots of coordinate vector differences
and moving them into an exponential. Addition theorems may also be used for
this purpose, and sometimes direct integration is even possible. Each method
has its strengths and weaknesses. An alternative integral transform to Fourier
transforms and Gaussian transforms is derived herein and utilized. A number of
consequent integrals of Macdonald functions, hypergeometric functions, and
Meijer G-functions with complicated arguments is given.
