Reducing a Class of Two-Dimensional Integrals to One-Dimension with an Application to Gaussian Transforms

Quantum theory is awash in multidimensional integrals that contain exponentials in the integration variables, their inverses, and inverse polynomials of those variables. The present paper introduces a means to reduce pairs of such integrals to one dimension when the integrand contains powers multiplied by an arbitrary function of <inline-formula><math display="inline"><semantics><mrow><mi>x</mi><mi>y</mi><mo>/</mo><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo></mrow></semantics></math></inline-formula> multiplying various combinations of exponentials. In some cases these exponentials arise directly from transition-amplitudes involving products of plane waves, hydrogenic wave functions, and Yukawa and/or Coulomb potentials. In other cases these exponentials arise from Gaussian transforms of such functions.