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In this paper we review the calculations that are needed to obtain thebosonic and fermionic effective potential at finite temperature and volume (atone loop). The calculations at finite volume correspond to $S^1\times R^d$topology. These calculations appear in the calculation of the Casimir energyand of the effective potential of extra dimensional theories. In the case offinite volume corrections we impose twisted boundary conditions and obtainsemi-analytic results. We mainly focus in the details and validity of theresults. The zeta function regularization method is used to regularize theinfinite summations. Also the dimensional regularization method is used inorder to renormalize the UV singularities of the integrations over momentumspace. The approximations and expansions are carried out within theperturbative limits. After the end of each section we briefly presentapplications associated to the calculations. Particularly the calculation ofthe effective potential at finite temperature for the Standard Model fields,the effective potential for warped and large extra dimensions and thetopological mass creation. In the end we discuss on the convergence andvalidity of one of the obtained semi-analytic results.

Report of the Detailed Calculation of the Effective Potential in Spacetimes with $S^1\times R^d$ Topology and at Finite Temperature