The authors consider equations of the form \[ iu_t(x,t)=\Delta_x u(x,t)+ W(x,t) u(x,t),\quad x\in\mathbb{R}^2,\;t> 0, \]
\[ u(x,0)= u_0(x), \] where \(W(x,t)\) and \(u_0(x)\) are compactly supported. The authors describe a fast algorithm for the imposition of the exact nonreflecting boundary conditions (ENRBC) on a unit disk. Analytical approach of the authors is classical, using the Fourier transform in space and the Laplace transform in time. Here the authors concentrate on the ENRBC for each Fourier mode, which is nonlocal in time and is shown to involve a convolution integral. The central problem the authors face is that the convolution kernel is not analytic and only its Laplace transform is known. The principal analytical result of this paper is that the Laplace transform of the convolution kernel can be approximated in the right half-plane with an absolute error by a sum of poles with the number of poles
\[ N(\varepsilon, \mu)\sim O(\log(1/\varepsilon)(\log\mu+ \log(1/\varepsilon))) \]
as \(\mu\to\infty\) and \(\varepsilon\to 0\).