  We study the zeta-function regularization of functional determinants of
Laplace and Dirac-type operators in two-dimensional Euclidean $AdS_2$ space.
More specifically, we consider the ratio of determinants between an operator in
the presence of background fields with circular symmetry and the free operator
in which the background fields are absent. By Fourier-transforming the angular
dependence, one obtains an infinite number of one-dimensional radial operators,
the determinants of which are easy to compute. The summation over modes is then
treated with care so as to guarantee that the result coincides with the
two-dimensional zeta-function formalism. The method relies on some well-known
techniques to compute functional determinants using contour integrals and the
construction of the Jost function from scattering theory. Our work generalizes
some known results in flat space. The extension to conformal $AdS_2$ geometries
is also considered. We provide two examples, one bosonic and one fermionic,
borrowed from the spectrum of fluctuations of the holographic $\frac{1}{4}$-BPS
latitude Wilson loop.
