lambertw, all branches

[Gil]

I looked your case k=-1 & k=1, x=-.1 + i ×1E-99.

With my initial guess=
'LN(((-.1,1.E-100)+1./e)*(e-\v/2.-1.)+\v/(((-.1,1.E-100)+1./e)*2./e)+1./e)'

= (-.11284787341,1.26802807724E-100),

the built-in multisolver from HP50 runs endless.

I looked in your programs, but could not understand which initial Xo formulae I should use for tiny imaginary part.

I tried, from what I guessed from your code, as a guess xo=ln(-a), and it worked.

When should I used that guess formula for my initial guess?

For example, xo=ln(-a) failed to produce any answer for
W0
(
(1.E-100,-.0000000001)
)
: a bad guess.

I had to use instead the first formulae to get an answer for
W0
(
(1.E-100,-.0000000001)
)
: (9.99905166733E-21,-1.00000000005E-10).

And what do you mean by I.log(-a)?
Is I.log(-x) a special function or is it i×ln(-x).

Could you help me on that matter?

Thanks in advance.