lambertw, all branches

[Gil]

1) You write
lua> a
(4.794621373315692+1.4183109273161312*I)
lua> I.W(a, 1, true)
(0+6.570796326794897*I)
(-0.033367069043767406+4.994921913968373*I)
(1.97521636071743e-005+5.000010560789135*I)
(3.753082249819371e-012+5.000000000009095*I)
(1.0141772755203484e-016+5*I)
(-6.474631941441244e-017+5*I)

We need more precisions to get real part right.

What do you mean by more precision?
Can we get better result, at least regarding the sign of the real part?[/b]

2)You give the examples for
"W0(-pi /2), that should be (0+i×pi/2)
"W-1(-pi /2), that should be (0 -i×pi/2).

Is there a way to get the real part almost = to zero (±1E-100)?

The answer is have an initial guess xo with real part = 0 —> not to have ln(-a)+t/2, but only t/2, as ln(-a) = ln(pi/2) = real.

Could we say, instead of that code line
xo= ln(-a) + (0 of A<. 5, else t/2)

IF a>=0
THEN xo= ln(-a) + (0 of A<. 5, else t/2)
ELSE xo= always t/2
END

Try W(k=-1)
For a=-2, and put zero for the real part of the "found xo initial guess. The iterating process finds all the same the good solution.
For any number a <0, for instance -0.4, and then procede as above, taking xo = t/2,and the expected answer seems to be always found!
Of course, it's not a mathematical solution, but a possible "trick answer" for dummies. To be thought over?