  Investigations are made on the saddle point calculations (SPC) under the
auxiliary field method in path integrations. Two different ways of SPC are
considered, Method(I) and Method(II), to be checked in an integral
representation of the Gamma function, \Gamma (N), as a bosonic example and in a
four-fermi type of Grassmann integral where one "fermion mass" \omega_0 differs
from the other N-degenerate species. The recipe of Method(I) seems rather
complicated than that of (II) superficially, but the case turns out to be
opposite in the actual situation. A general formalism allows us to calculate
for \Gamma (N) up to O(1/N^{14}). It is found that both happen to coincide in
the bosonic case but in the fermionic case Method(II) shows a huge deviation in
the weak coupling region where \omega_0 \ll 1.
