using Plots, ComplexPhasePortrait, ApproxFun, SingularIntegralEquations
gr();

ε = 0.05
x = 0.1
Γ = Segment(-1 , x-ε) ∪ Segment(x+ε , 1)
t = Fun(Γ)
f = Fun(1/(t - x))

@show log(1 - x) - log(1+x)

plot(f; label="f", title="eps = $(ε), integral = $(sum(f))",ylims=(-40,40))
scatter!([x],[0.0];label="x")

x = Fun(-1 .. 1)
f = exp(x)*sqrt(1-x^2)
plot(f; label="function")
plot!(hilbert(f); label="hilbert transform")

x = Fun(-1 .. 1)
f = exp(x)/sqrt(1-x^2)
plot(f; label="function", ylims=(-5,5))
plot!(hilbert(f); label="hilbert transform")

f = exp(x)
xx = range(-0.9999; stop=0.9999, length=500)
plot(f; ylims=(-5,5), label="function")
plot!(xx, hilbert.(f, xx); label="hilbert transform")

f = exp(x)sqrt(1-x^2)
cauchy(f, 0.1+0.0im)+cauchy(f, 0.1-0.0im) , -im*hilbert(f)(0.1)

x = Fun()
norm(hilbert(1/sqrt(1-x^2)))

plot(hilbert(sqrt(1-x^2)))

hilbert(Fun(1,Legendre()), 0.1), (log(1-0.1)-log(0.1+1))/π

f = exp(x)sqrt(1-x^2)
cauchy(f, 10.0+im), cauchy(f, 10.0-im)

f(0.1), 2real(cauchy(f, 0.1+0.0im)), -2real(cauchy(f, 0.1-0.0im))

-hilbert(f, 0.1), 2imag(cauchy(f, 0.1+0.0im)), 2imag(cauchy(f, 0.1-0.0im))