Excluding at first zero modes, we evaluate formula (17) for ratios
of functional determinants. The temporal integral on the right-hand side can
be performed efficiently following Ref. [10]. Here we present an even
more direct method, by which we express the result in terms of solutions of
Gelfand-Yaglom's initial-value problem for Dirichlet boundary conditions,
and of a dual problem for periodic and antiperiodic boundary conditions.
Dirichlet Case
The Gelfand-Yaglom initial-value problem consists in the search for a
function 
solving the following equations:
By differentiating these three equations with respect to the parameter g,
we obtain for 
the inhomogeneous
initial-value problem
The unique solution of equations (41) can easily be expressed in
terms of our arbitrary set of solutions 
and 
as
follows
thus leading to
In terms of the same functions, the general solution of the inhomogeneous
initial-value problem (42) can be seen to have the form
Comparison with (34) shows that at the final point 
which together with (44) implies the following simple relation for
the Green function (34) with Dirichlet's boundary conditions:
Inserting this into (17), we find for the ratio of functional
determinants the simple formula
The constant of integration is fixed by applying (48) to the trivial
case g=0, where 
and the solution to the initial-value
problem (41) is
At g=0, the left-hand side of (50) is unity, determining 
and the final result for g=1:
This compact formula was first derived by Gelfand and Yaglom [2] via a
direct calculation of the determinant arising in a time-sliced path
integrals [1].
Periodic and Antiperiodic Case
Our technique makes it straight-forward to derive an equally compact formula
for periodic and antiperiodic boundary conditions. For this purpose we
introduce another homogeneous initial-value problem whose boundary
conditions are dual to Gelfand and Yaglom's in (41):
In terms of the previous arbitrary set and of
solutions of the homogeneous differential equation, the unique solution of (51) reads
This can be combined with the time derivative of (43) at to
yield
By differentiating Eqs. (51) with respect to g, we obtain the
following inhomogeneous initial-value problem for 
:
whose general solution reads in analogy to (45)
where the dot denotes the time derivative with respect of the first argument
of 
. With the help of identities (26) and (27), the combination 
at
can now be expressed in terms of the periodic and antiperiodic
Green functions (40), in analogy to (46),
Together with (53), this yields for the temporal integral on the
right-hand sides of (16) and (17) the simple expression
analogous to (47)
This is inserted into formula Eq. (17) yielding for periodic and
antiperiodic boundary conditions
where 
. The constant of
integration C is fixed in the way described after Eq. (17). We go
to g=1 and set 
. For the operator 
, we can easily solve the
Gelfand-Yaglom initial-value problem (41) as well as the dual one (51) by
so that (58) determines C by
Hence we obtain the final results for periodic boundary conditions
and for antiperiodic boundary conditions
The intermediate expressions in (50), (61), and (62)
show that the ratios of functional determinants are ordinary determinants of
two arbitrary independent solutions 
and 
of the
homogeneous differential equation 
. As such, the results are manifestly invariant under arbitrary
linear transformations of these functions 
.