  We revisit the computation of (2-modified) Fredholm determinants for
operators with matrix-valued semi-separable integral kernels. The latter occur,
for instance, in the form of Green's functions associated with closed ordinary
differential operators on arbitrary intervals on the real line. Our approach
determines the (2-modified) Fredholm determinants in terms of solutions of
closely associated Volterra integral equations, and as a result offers a
natural way to compute such determinants.
  We illustrate our approach by identifying classical objects such as the Jost
function for half-line Schr\"odinger operators and the inverse transmission
coefficient for Schr\"odinger operators on the real line as Fredholm
determinants, and rederiving the well-known expressions for them in due course.
We also apply our formalism to Floquet theory of Schr\"odinger operators, and
upon identifying the connection between the Floquet discriminant and underlying
Fredholm determinants, we derive new representations of the Floquet
discriminant.
  Finally, we rederive the explicit formula for the 2-modified Fredholm
determinant corresponding to a convolution integral operator, whose kernel is
associated with a symbol given by a rational function, in a straghtforward
manner. This determinant formula represents a Wiener-Hopf analog of Day's
formula for the determinant associated with finite Toeplitz matrices generated
by the Laurent expansion of a rational function.
