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SUMMARY:Nick Trefethen (University of Oxford)
DTSTART:20201110T140000Z
DTEND:20201110T150000Z
DTSTAMP:20260422T201413Z
UID:CAvid/20
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CAvid/20/">A
 pproximation on complex domains and Riemann surfaces</a>\nby Nick Trefethe
 n (University of Oxford) as part of CAvid: Complex Analysis video seminar\
 n\nLecture held in N/A.\n\nAbstract\nLet f be a function analytic on a clo
 sed Jordan region E apart\nfrom a finite number of branch point singularit
 ies on the boundary.\nWe show how f can be approximated by rational functi
 ons on E with\nroot-exponential convergence\, i.e.\, errors $O(\\exp(-C \\
 sqrt n))$ with\n$C>0$.  Such approximations lead to "lightning solvers" fo
 r Laplace\nproblems in planar domains.  Then we move to "reciprocal-log" o
 r\n"log-lightning" approximations involving terms of the form\n$c/(\\log(z
 -z_k) - s_k)$.  Now one gets exponential-minus-log convergence\,\ni.e.\, $
 O(\\exp(-C n/\\log n))$.  Moreover\, the reciprocal-log functions\ncan be 
 analytically continued around the branch points to provide\napproximation 
 on further Riemann sheets.  This work (with Yuji\nNakatsukasa) is very new
 \, and there are many open questions.\n
LOCATION:https://researchseminars.org/talk/CAvid/20/
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