Approximation on complex domains and Riemann surfaces

Abstract: Let f be a function analytic on a closed Jordan region E apart from a finite number of branch point singularities on the boundary. We show how f can be approximated by rational functions on E with root-exponential convergence, i.e., errors $O(\exp(-C \sqrt n))$ with $C>0$. Such approximations lead to "lightning solvers" for Laplace problems in planar domains. Then we move to "reciprocal-log" or "log-lightning" approximations involving terms of the form $c/(\log(z-z_k) - s_k)$. Now one gets exponential-minus-log convergence, i.e., $O(\exp(-C n/\log n))$. Moreover, the reciprocal-log functions can be analytically continued around the branch points to provide approximation on further Riemann sheets. This work (with Yuji Nakatsukasa) is very new, and there are many open questions.

complex variablesdynamical systemsnumerical analysis

Audience: researchers in the topic

CAvid: Complex Analysis video seminar

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