Conical functions of purely imaginary order and argument

Associated Legendre functions are studied for the case where the degree is in conical form −½ + iτ (τ real), and the order iμ and argument ix are purely imaginary (μ and x real). Conical functions in this form have applications to Fourier expansions of the eigenfunctions on a closed geodesic. Real-valued numerically satisfactory solutions are introduced which are continuous for all real x. Uniform asymptotic approximations and expansions are then derived for the cases where one or both of μ and τ are large; these results (which involve elementary, Airy, Bessel and parabolic cylinder functions) are uniformly valid for unbounded x.

REPORT ON THE DETAILED CALCULATION OF THE EFFECTIVE POTENTIAL IN SPACETIMES WITH S1 × Rd TOPOLOGY AND AT FINITE TEMPERATURE

In this paper, we review the calculations that are needed to obtain the bosonic and fermionic effective potential at finite temperature and volume (at one loop). The calculations at finite volume correspond to S1 × Rd topology. These calculations appear in the calculation of the Casimir energy and of the effective potential of extra dimensional theories. In the case of finite volume corrections, we impose twisted boundary conditions and obtain semi-analytic results. We mainly focus in the details and validity of the results. The zeta function regularization method is used to regularize the infinite summations. Also the dimensional regularization method is used in order to renormalize the UV singularities of the integrations over momentum space. The approximations and expansions are carried out within the perturbative limits. After the end of each section, we briefly present applications associated to the calculations. Particularly the calculation of the effective potential at finite temperature for the standard model fields, the effective potential for warped and large extra dimensions, and the topological mass creation. In the end, we discuss on the convergence and validity of one of the obtained semi-analytic results.