using Plots, ComplexPhasePortrait, ApproxFun
gr();

f = x -> x^2/(x^4+1)
phaseplot(-3..3, -2..2, f)

z₁,z₂,z₃,z₄ = exp(im*π/4), exp(3im*π/4), exp(5im*π/4), exp(7im*π/4)

res₁ = z₁^2 / ((z₁ - z₂)*(z₁ - z₃)*(z₁ - z₄) )
res₂ = z₂^2 / ((z₂ - z₁)*(z₂ - z₃)*(z₂ - z₄) )

2π*im*(res₁ + res₂), sum(Fun(f, Line()))

res₃ = z₃^2 / ((z₃ - z₁)*(z₃ - z₂)*(z₃ - z₄) )
res₄ = z₄^2 / ((z₄ - z₁)*(z₄ - z₃)*(z₄ - z₂) )

-2π*im*(res₃ + res₄), sum(Fun(f, Line()))

f = x -> x^2/(x+im)^3
z = 2.0+2.0im
sum(Fun(x-> f(x)/(x - z), Line()))/(2π*im) - f(z)

f = x -> x^2/(x+im)^3
z = 2.0-2.0im
sum(Fun(x-> f(x)/(x - z), Line()))/(2π*im) , f(z)

f = x -> x^2/(x-im)^3
z = 2.0-2.0im
-sum(Fun(x-> f(x)/(x - z), Line()))/(2π*im) , f(z)

f = x -> exp(im*x)/(x+im)
z = 2 + 2im
sum(Fun(x-> f(x)/(x - z), -500 .. 500))/(2π*im) - f(z)

xx = -200:0.1:200
plot(xx,real.(f.(xx)))

z = -2-im
f = x -> exp(im*x)/(x+im)
sum(Fun(x-> f(x)/(x - z), -500 .. 500))/(2π*im)

z = -2-im
f = x -> exp(im*x)/(x-im)
sum(Fun(x-> f(x)/(x - z), -500 .. 500))/(2π*im), f(z)

z = -2-im
f = x -> exp(-im*x)/(x-im)
-sum(Fun(x-> f(x)/(x - z), -500 .. 500))/(2π*im), f(z)

θ = range(0; stop=π/2, length=100)
plot(θ, sin.(θ); label="sin t")
plot!(θ, 2θ/π; label = "2t / pi")

f = x -> exp(im*x)*x/(x^2+1)
sum(abs.(Fun(f, 0 .. 2000)))

f = x -> exp(im*x)*x/(x^2+1)

sum(Fun(f, -30000 .. 30000))

2π*im*exp(-1)*im/(im+im)  # 2π*im* residue of g(z)exp(im*z) at z = im