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

t = Fun(0.0 .. 40)
f = exp(-t)
fourier(f, -2.0), im/(im-2.0)

1-(fourier(-exp(-t), -2.0) + fourier(-exp(Fun(-40 .. 0)), -2.0)), (2.0^2 + 3)/(2.0^2+1)

κ = z -> imag(z) > 0 ? (z+im*sqrt(3))/(z+im) :
                       (z-im)/(z-im*sqrt(3))

κ(0.1+eps()im) - κ(0.1-eps()im)*(3+0.1^2)/(1+0.1^2)

Y = z -> imag(z) > 0 ? -im*(1-sqrt(3))/(1+sqrt(3))/(z+im*sqrt(3)) :
                       -2im/((z-im)*(1+sqrt(3)))

s = 0.1

Y(s+eps()im) - Y(s-eps()im)

im/(s-im)*(s+im)/(s + im*sqrt(3))

φ =  z -> imag(z) > 0 ? -im*(1-sqrt(3))/(1+sqrt(3))/(z+im) :
                       -2im/((z-sqrt(3)*im)*(1+sqrt(3)))

g = s -> (s^2+3)/(s^2+1)

s = 0.1
φ(s+eps()im) - φ(s-eps()im)*g(s) - im/(s-im)

t = Fun(0 .. 10)

u = 2exp(-sqrt(3)*t)/(1+sqrt(3))
x = 0.1

u(x) + sum(exp(-abs(t-x))*u) - exp(-x)