Abstract
Occupation number representation of the twoβdimensional harmonic oscillator and some operator formulae are used in a simple algebraic derivation of complicated integrals. The calculation of full oscillatorβ and radial integrals of rΜ^w^ and exp (\documentclass{article}\pagestyle{empty}\begin{document}$ (iw\hat{\varphi})$\end{document}), where w is an arbitrary positive or negative integer, are performed by an integral transform, leading to a generalized Gauss matrix element. Thus it is possible, because of the back transformation, to derive from one generalized Gauss matrix element all matrix elements which are permitted by the selection rules. Some integrals of rΜ^w^ and exp (\documentclass{article}\pagestyle{empty}\begin{document}$ (iw\hat{\varphi})$\end{document}), Laguerre polynomials, and Bessel functions are completely new. For the already known integrals, the mathematical labour is considerably reduced. The relation between the twoβdimensional 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.