<p>We write down explicit algebraic formulas for the Szeg\H{o}, Garabedian and Bergman kernels for specific two-connected planar domains. We use these results to derive integral representations for a biholomorphic invariant relating the Bergman and Szeg\H{o} kernels. We use the formulas to study the asymptotic behavior of these kernels as a family of two-connected domains approaches the unit disc. We derive an explicit formula for the Green's function for the Laplacian for special values on two-connected domains. Every two-connected domain is biholomorphic to a unique two-connected domain of the type we consider. This allows one to write down formulas for the kernel functions on a general two-connected domain.</p>