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

x = Fun()
w = 1/sqrt(1-x^2)
z = 0.1+0.1im

n = 10
L = zeros(ComplexF64,n,n)
L[1,1] = 1
L[2,1] = -z
L[2,2] = 1
for k=3:n
    L[k,k-1] = -z
    L[k,k-2] = L[k,k] = 1/2
end



C = L \ [ im/(2sqrt(z-1)sqrt(z+1)); 1/(2im); zeros(n-2)]

T₅ = Fun(Chebyshev(), [zeros(5);1])
cauchy(T₅*w,z) - C[6]

x = Fun()
w = 1/sqrt(1-x^2)
z = 5+6im

n = 100
L = zeros(ComplexF64,n,n)
L[1,1] = 1
L[2,1] = -z
L[2,2] = 1
for k=3:n
    L[k,k-1] = -z
    L[k,k-2] = L[k,k] = 1/2
end

C = L \ [ im/(2sqrt(z-1)sqrt(z+1)); 1/(2im); zeros(n-2)]

T₂₀ = Fun(Chebyshev(), [zeros(20);1])

C[21], cauchy(T₂₀*w, z)

L[2:end,1:end-1]

C = L[2:end,1:end-1]\ [1/(2im); zeros(n-2)]

C[6]- cauchy(T₅*w, z)

x = 0.1

T = Fun(Chebyshev(),[zeros(n);1])
hilbert(w*T,x) - Fun(Ultraspherical(1), [zeros(n-1);1])(x)