using Plots, ComplexPhasePortrait, ApproxFun
gr();

z = -2 + 1.0im
phaseplot(-3..3, -3..3, ζ -> 1/ζ)
t = 0:0.1:1
γ = 1 .+ (z-1)*t
plot!(real.(γ), imag.(γ); color=:black, label="contour", arrow=true)

phaseplot(-3..3, -3..3, z -> log(z))

log(-1+0.0im), log(-1-0.0im)

θ = range(0, stop=0.9π, length=100)
a = exp.(im*θ)

z = exp(0.9*π*im)

γ = 1 .+ (z-1)*t
phaseplot(-3..3, -3..3, ζ -> 1/ζ)
plot!(real.(γ), imag.(γ); color=:black, label="contour", arrow=true)
plot!(real.(a), imag.(a), color=:black, linestyle=:dash, label="deformed", arrow=true)

sum(Fun(ζ -> 1/ζ, Arc(0.,1.,(0.,π)))),
sum(Fun(ζ -> 1/ζ, Arc(0.,1.,(0.,-π))))

log(-1.0+0im) - log(-1.0-0im), sum(Fun(ζ -> 1/ζ, Circle()))

phaseplot(-3..3, -3..3, z -> log(z))
z = -2 + 2im

γ = (z,t) -> 1 + t*(z-1)
tt = range(0,stop=1,length=100)

plot!(real.(γ.(z,tt)), imag.(γ.(z,tt)); color=:black, label="contour", arrow=true)
plot!(real.(1 ./ γ.(z,tt)), imag.(1 ./ γ.(z,tt)); color=:black, linestyle=:dash, arrow=true, label="change of variables contour")
plot!(real.(γ.(1/z,tt)), imag.(γ.(1/z,tt)); color=:black, linestyle=:dashdot, arrow=true, label="deformed contour")

phaseplot(-3..3, -3..3, z -> -log(1/z))

α = 0.5
phaseplot(-3..3, -3..3, z -> z^α)

sqrt(-1-0.0im) , -sqrt(-1 + 0.0im)

x = -2
(x+0.0im)^α - (x-0.0im)^α, 2im*(-x)^α*sin(π*α)

x = -1
(x+0.0im)^α + (-1-0.0im)^α, 2*(-x)^α*cos(π*α)

sqrt(x+0.0im) - sqrt(x-0.0im) == 2im*sqrt(-x)

sqrt(x+0.0im) + sqrt(x-0.0im)

φ = z -> sqrt(z-1)*sqrt(z+1)
phaseplot(-3..3, -3..3, φ)

x = 0.1
φ(x + 0.0im) + φ(x - 0.0im)

x = 0.1
φ(x + 0.0im) - φ(x - 0.0im), 2im*sqrt(1-x^2)

φ = z -> sqrt(z-1)sqrt(z+1)

x = -3.0
φ(x+0.0im), φ(x-0.0im)

z = 0.2+0.2im
x = 0.1
ε = 0.001
scatter([x],[0.]; label="x", xlims=(-1.5,1.5), ylims=(-0.5,0.5))
scatter!([real(z)],[imag(z)]; label="z")
plot!(-1..1; label="Unit interval", linestyle=:dot)

Γₓ =  Arc(z, 0.1, (-π/2,π)) ∪ Segment(0.2+0.1im,0.2 +0.0im) ∪ Segment(0.2  +0.0im, -0.9 +0.0im) ∪
     Circle(-1.0, 0.1) ∪ Segment(-0.9 -0.0im, 0.2  -0.0im) ∪ Segment(0.2-0.0im, 0.2 - 0.2im) ∪
      Segment(0.2 - 0.2im, -1.2-0.2im) ∪ Segment(-1.2 -0.2im, -1.2+ 0.2im) ∪ 
        Segment(-1.2+ 0.2im, 0.1+0.2im)

plot!(Γₓ; label="Contour")

z = 0.2+0.2im
x = 0.1
ε = 0.001
scatter([x],[0.]; label="x", xlims=(-1.5,1.5), ylims=(-0.5,0.5))
scatter!([real(z)],[imag(z)]; label="z")
plot!(-1..1; label="Unit interval", linestyle=:dot)

Γ̃ₓ =  Arc(z, 0.1, (-π/2,π)) ∪ Segment(0.2+0.1im,0.2 -0.2im) ∪    Circle(-1.0, 0.1) ∪
      Segment(0.2 - 0.2im, -1.2-0.2im) ∪ Segment(-1.2 -0.2im, -1.2+ 0.2im) ∪ 
        Segment(-1.2+ 0.2im, 0.1+0.2im)

plot!(Γ̃ₓ; label="Contour")

z₀ = 0.2 +0.2im
z = 0.4 +0.2im

scatter([real(z₀)],[imag(z₀)]; label="z_0")
scatter!([real(z)],[imag(z)]; label="z")
plot!(Circle(z₀, 0.1); label="gamma_r")
plot!(Circle(z₀, 0.3); label="gamma_R")

κ = z -> 1/(sqrt(z-1)sqrt(z+1))
phaseplot(-3..3, -3..3, κ)

x = 0.1
κ(x + 0.0im) - κ(x - 0.0im) , -2im/sqrt(1-x^2)

x = -2.3
κ(x + 0.0im) - κ(x - 0.0im) 

μ = z -> (log(z-1) - log(z+1))/(2π*im)
phaseplot(-3..3, -3..3, μ)

x = 0.3
μ(x + 0.0im) - μ(x - 0.0im) 

x = -4.3
μ(x + 0.0im) - μ(x - 0.0im)