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Riemann–Hilbert problems and Stokes phenomena

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ARA - Applicable resurgent asymptotics: towards a universal theory

Riemann–Hilbert problems consist of recovering a piecewise analytic function from information about jumps along branch cuts. To quote Wikipedia (as of today): “Stokes phenomenon, discovered by G. G. Stokes (1847, 1858), is that the asymptotic behaviour of functions can differ in different regions of the complex plane”. We demonstrate how Stokes phenomena can lead naturally to a Riemann–Hilbert problem which in fact uniquely determines the analytic function, beginning, of course, with everyone’s favourite example: the Airy function. We further review the applications to Painlevé equations, and finally show that integrals with coalescing saddles can also be reformulated as a Riemann–Hilbert problem, in a way that, perhaps, avoids the computational pitfalls of applying quadrature directly to integral reformulations along steepest descent contours

This talk is part of the Isaac Newton Institute Seminar Series series.

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