Conformal mapping methods for spectral zeta function calculations.

We first show how to relate two spectral zeta functions corresponding to conformally equivalent two-dimensional smooth Riemannian manifolds. Next, the functional determinant of the Laplacian on an annulus is used to calculate the functional determinant of the Laplacian on a region bounded by two ellipses. We develop perturbation theory for Hermitian partial differential operators and show how this, combined with a conformal map from a disk to an elliptic region, can be used to derive a perturbative expansion for the spectral zeta function of the Laplacian on an elliptic region that is nearly circular. Finally, this perturbative expansion of the zeta function is used to approximate quantities of interest such as the functional determinant and heat kernel coefficients.