Properties of Fermi and symmetrized Fermi functions and applications in nuclear physics

The aim of this article is to survey advances made in investigating the pro perties of the Fermi (F) and symmetrized Fermi (SF) functions and in using them as approximants for basic physical inputs in various applications in n uclear physics and related areas, such as the physics of hypernuclei and me tal clusters. The evaluation of the F- and SF-type integrals, also taking i nto account more general limits, is considered on the basis of either the S ommerfeld approximation, or beyond that, when, e.g., rapidly oscillating fu nctions are involved in the integrand. Particular attention is paid to the "small exponential terms" and such topics as the Fourier and Bessel transfo rms of the F and SF functions, analytic properties, and the Dingle represen tation of the F function. Applications refer to nuclear diffraction in the scattering of particles by nuclei, generalized expressions of the harmonic oscillator energy level spacing for its variation with the particle number, and the study of the Woods-Saxon-type potentials and their use in problems of hypernuclei and metal clusters.