lambertw, all branches

[Albert Chan]

Hi, Gil

Complex Conjugate formula is always true, if we have signed zero.
With signed-zero, arg(z) + arg(conj(z)) = 0

Let z = r * cis(θ)      // polar form, θ = arg(z)

ln(conj(z)) = ln(r*cis(-θ)) = ln(r) - θ*I = conj(ln(r) + θ*I) = conj(ln(z))

Read from right to left, conj can go inside ln

x = W_k(a)
x + ln(x) = ln(a) + 2*k*pi*I
conj(x + ln(x)) = conj(ln(a) + 2*k*pi*I)
conj(x) + ln(conj(x)) = ln(conj(a)) + 2*(-k)*pi*I
conj(x) = W_-k(conj(a))

--> W_k(a) = conj(W_-k(conj(a)))