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

An integral transform for quantum amplitudes