lambertw, all branches

[Gil]

A question, but sorry I do not understand how to interpret your code:

"x + ln(x) = ln(|a|) + T
Im(x + ln(x)) = T/i = [pi/2, pi]

a ≈ -1 → guess x = T/2 = [pi/4, pi/2]*i

We could hide log1p singularity inside branch region, for simpler and better W0 guess
Code:
< if k==0 then x = I.abs(a+1)<.6 and T/2 or I.log1p(a) end
> if k==0 then x = I.log1p(2*a)/2 end"

I take x=abs (a+1),
A)"if that result is less than 0.6"
B) A)"if that result is less than 0.6 AND that result is also less than a×EXP(a)
C) About T/2: it is an interval from [pi/4 pi/2]×pi, yes?
My question, how can I say that abs(a) is less than T/2 = an imaginary number with no real part?
D) "if the result abs(a) is not less than 0.6, then xo=I.log1p(2*a)/2

Sorry for these questions, but I am not a math guy, as you know, that however wishes to partially understand the criteria used in order to have his program work in any situation.

Thanks again.