Lambert function and Wolfram or "±infinity+i×K=±infinity" ?

[Albert Chan]

Update with generalized W formula. Set θ=pi, we have W0(-∞) = ∞ + pi*I

Let z = ∞*cis(θ), r = ln(|z|)

We want to show w = W₀(z) = ln(z) = r + θ*I
By checking for round-trip, i.e. w * e^w = z

w = r + θ*I ≈ r * cis(θ/r)                     // polar form
e^w = e^(r + θ*I) = e^r * cis(θ)

w * e^w ≈ r*e^r * cis(θ/r + θ)

|z| → ∞ --> r → ∞ too

w * e^w = ∞ * cis(0 + θ) = z

From definition of k-th branch of W

k = (w + ln(w) - ln(z)) / (2*pi*i)
2*pi*k = im(w + ln(w) - ln(z)) = θ + atan(θ/r) - θ = 0

--> k = 0
--> W₀(∞*cis(θ)) = ln(∞*cis(θ)) = ∞ + θ*I

Again, from definition of k-th branch:

--> W_k(∞*cis(θ)) = ln(∞*cis(θ)) + 2*k*pi*I