lambertw, all branches

[Gil]

In your second post, Albert, you wrote

RE: lambertw, all branches
f = x + ln(x) - ln(a) - 2*k*pi*I

Solve for f=0 is easy if x is big.
For |k| > 1, we can solve directly, since x imaginery part ≈ 2*k*pi (see OP example)

For k=-1, solve for f=0 is especially difficult.
Fortunately, we can flip it, to solve for k=1 instead.
W(z, k) == conj(W(conj(z), -k)
In fact, I flip all negative k's. So now, k is non-negative.

But W-1(-1E-330)=
-697.32277629546016099540752740546566360568199201428307824605649289390557552642032068023​149199134240178698554915132

And W1(-1E-330)=
-697.32281706295494653348757314785952050772713371449039168 +6.2922084418351405256893689216267218827742283825238692107 i

It seems I missed a point when I may use
W(z, k) == conj(W(conj(z), -k).

Thanks for your lights, Albert.