(28/48/50) Lambert W Function

[Albert Chan]

(04-03-2023 07:24 PM)John Keith Wrote:  My program gets exact or near-exact results for all branches, but it is slower for branches other than 0 and -1.

I was getting stupid. Of course you get near-exact result for other branches.

For branches besides (0, -1), a ≈ -1/e is not that important, since W(-1/e) ≠ -1
With huge complex parts in the way, there is no catastrophic cancellation issues.      (*)

The singularity is at only at 0, not -1/e

W_-1 is the "hard" branch, with singularities at both places, 0 and -1/e.

Plot[{Re[LambertW[-1,x]], Im[LambertW[-1,x]]}, {x,-1,1}]


(*) Except for W₁: lambertw(z, k) == conj(lambertw(conj(z), -k))

>>> lambertw(-.36787944+1e-99j, k=-1)
mpc(real='-1.0000798057615958', imag='-3.4063941215654215e-95')

>>> conj(lambertw(-.36787944-1e-99j, k=1))
mpc(real='-1.0000798057615958', imag='-3.4063941215654215e-95')