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

[Albert Chan]

(01-16-2024 06:11 PM)Gil Wrote:  Instead, the answer seems rather to be (infinity+i*pi).
Then (infinity+i*pi) × exp(infinity) × cos (pi) =
Infinity × -1= -infinity.

(∞ + pi*I) * -∞ = -∞ - ∞*I

Still does not get back -∞, unless ... (∞ + pi*I) = ∞ ?

(∞ + pi*I) * -∞ = ∞ * -∞ = -∞

But then, we are just picking and choosing!

We start with saying W(-∞) = ∞ + pi*I ≠ ∞
But to make it roundtrip, we had to simplify (∞ + pi*I) = ∞

We may turn it around, and say ∞ = ∞ + (any finite number, real or complex)

∞ * e^∞ = ∞ * e^(∞+θ*i) = ∞ * e^∞ * cis(θ) = ∞ * cis(θ)

--> W(∞ * cis(θ)) = ∞      // infinity is weird!