logkernel sometimes throws domain error due to acos

[dlfivefifty]

SingularIntegralEquations.jl/src/singfuncauchy.jl

Line 170 in 1e4cc41

y = r*exp(im*acos(x/r))

logkernel(1/sqrt(1-x^2),-5.0)

[dlfivefifty]

I think the code needs to be redesigned anyways using the OP point of view.

Annoyingly, you can't work around this by perturbing -5.0.

[MikaelSlevinsky]

The problem is a rounding error is causing the real part of the Joukowsky inverse to be larger in magnitude than the absolute value by one bit.

Since x and y are both needed, a fix would use reim(joukowskyinverse(...)), though this might be less efficient.

[dlfivefifty]

Hmm, why don't we just simplify r*exp(im*acos(x/r)) as r + im*sqrt(r^2-x^2) or similar?

[dlfivefifty]

PS Realized it's very easy to calculate equilibrium measures in the complex plane:

ε = 0.1
    x = real(Fun(Segment(0,1))) ; y = imag(Fun(Segment(ε*im,im)))
    w(a,b) = x^a*(1-x)^b + (y-ε)^b*(1-y)^a
    μ = SingularIntegral(space(w(-0.5,-0.5)),0) \ ones(domain(w(-0.5,-0.5)))

Unfortunately it doesn't have the nice property that real-line EMs do that logkernel(μ*x^k) is a degree k-1 polynomial...