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A method has been developed for computing the gravitational force field of an axially symmetric flat galaxy from its surface mass density. The method is simple to program, fast, and accurate. An inversion formula is derived that allows computation of surface density from rotation curves by use of any method that converts density to force. The method is compared with a method of Clutton-Brock that utilizes Hankel transforms of Laguerre functions.

Occupation number representation of the two-dimensional harmonic oscillator and some operator formulae are used in a simple algebraic derivation of complicated integrals. The calculation of full oscillator- and radial integrals of rw and exp (), where w is an arbitrary positive or negative integer, are performed by an integral transform, leading to a generalized Gauss matrix element. Thus it is possible, because of the back transformation, to derive from one generalized Gauss matrix element all matrix elements which are permitted by the selection rules. Some integrals of rw and exp (), Laguerre polynomials, and Bessel functions are completely new. For the already known integrals, the mathematical labour is considerably reduced. The relation between the two-dimensional oscillator and the hydrogen atom and their angular momentum properties are discussed. A survey on the various methods applied to the oscillator problem, from complex integration to noncompact Lie groups, and a comprehensive bibliography on this important spectroscopic field are given.

Using the generating function for Hermite polynomialsH n two integrals developed in computation of the bond energy of atomic nucleus have been solved. Furthermore the probability distribution of the coordinate of quantum harmonic oscillator has been derived with the aid of suitable integral representation ofH n(x). Presented computations are simpler then usually exploited recurrence equation technique.

Using the equivalent-quadrature'' relation between diagonalization of the s-wave kinetic energy H/sup o/ in a Laguerre-type basis and Chebyschev quadrature of the second kind, a procedure is given by which potential-scattering phase shifts may be constructed from approximations to the Fredholm determinant constructed using only square-integrable (L/sup 2/) functions. For the problem of electron-hydrogen scattering in the static approximation, highly accurate phase shifts are obtained over a continuous range of energies from a single major computational step. (auth)