022 Nick Trefethen
022 Nick Trefethen
Nick Trefethen (University of Oxford, UK)
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Approximation on complex domains and Riemann surfaces
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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.
Rod Halburd | |
193 | |
11/17/2020 | |
01:02:09 | |
CAvid, Complex Analysis |