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

[Albert Chan]

I have extended proof for W0(-∞) = ∞ + pi*I, for generalized case

(01-17-2024 02:06 AM)Albert Chan Wrote:  ...
--> 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

(01-17-2024 03:07 PM)Gil Wrote:  Cases z=a+ib, a=±inf, b=±inf
...

Unless z is in polar form, we don't know θ. (∞/∞ is indeterminate form)
mpmath, with slope=nan, θ = atan2(b,a) = nan

p2> log(mpc(inf,inf))
(+inf + nanj)

Comment: Technically we do know θ. Above example, 0 ≤ θ ≤ pi/2.
IEEE 754-2008 standard pick the mid-point for θ, instead of nan.

lua> I.log(I.new(inf,inf))
(inf+0.7853981633974483*I)

p3> from cmath import *
p3> log(complex(inf,inf))
(inf+0.7853981633974483j)