  We develop a new formula called a mode recombination formula, and we can
recast the effective potential at finite temperature in one-loop approximation
for fermion and scalar fields on the $D$-dimensional spacetime, $S_{\tau}^1
\times R^{D-(p+1)}\times\prod_{i=1}^pS_i^1$ into a convenient form for
discussing nonanalytic terms, which cannot be written in the form of any
positive integer power of the field-dependent mass squared, in the effective
potential. The formula holds irrespective of whether the field is a fermion or
a scalar and of boundary conditions for spatial $S_i^1$ directions and
clarifies the importance of zero modes in the Matsubara and Kaluza-Klein modes
for the existence of the nonanalytic terms. The effective potential is
drastically simplified further to obtain the nonanalytic terms in easier and
more transparent way. In addition to reproducing previous results, we find that
there exists no nonanalytic term for the fermion field with arbitrary boundary
condition for the spatial $S_i^1$ direction, which is also the case for the
scalar field with the antiperiodic boundary condition for the spatial
direction.
