Complex scaling – analytic continuation of Inclusions

[jagot]

A method for implementing absorbing boundary conditions is complex scaling, where the coordinate is rotated into the complex plane such that the plane waves exp(im*k*x) become exponentially damped. This is accomplished by simply evaluating all functions on the complex contour instead of the original domain (provided they can be analytically continued into the complex plane, of course). This does not work with the Inclusion machinery, since all indexing operations are checked and the domain given by the axes of the basis naturally does not include the complex plane.

What would be a nice way to implement this?

[jagot]

I suppose the mathematically correct solution would be for axes(B,1) to return the complex contour itself, although I do see that there could be numerical issues checking whether a complex number falls on the contour or not. I.e., if the contour is a 6° degree ray, then (0.9945218953682733, 0.10452846326765347) is on that ray, but how to check that?

Scratch that, better to say "this function is defined on this region of the complex plane (e.g. the unit circle)". I guess this is related to #10 then.

[dlfivefifty]

This has been discussed a lot in DomainSets.jl. I think what we want is "fuzzy domains" where the tolerance is built into the domain. So for example if the axis is exp.(Inclusion(0..1.0)) it will literally only contain the ComplexF64s such that z === exp(im*x) for some x isa Float64. If you want to allow for error you need to be precise, and say use a map of a complex rectangle, perhaps something like:

ε = 1E-10 # tolerance
x = Inclusion((-ε..1+ε) .+ im.*(-ε..ε)) # fuzzy interval
exp.(x) # fuzzy arc

[dlfivefifty]

One possibility is to use inbounds_getindex to mean analytic continuation...