lambertw, all branches

[Albert Chan]

I am starting a new thread, to discuss issues of solving W(a, k=0), extended to complex numbers.

The thread is based on this post, which supposed guaranteed correct branch, if f=0

(04-02-2023 11:12 PM)Albert Chan Wrote:  

(03-26-2023 06:43 PM)Albert Chan Wrote:  Because solving for y = e^W(a) is easier, W code is not recommended.

W code is perfect for other branches! (expW code for W branch 0, -1)

Numbering the branches of the Lambert W function



Except for 2*k*pi*I term, it is basically same formula for W code!

lua> f = fn'x: x + I.log(x) - I.log(a) - 2*k*pi*I'
lua> df = fn'x: 1 + 1/x'

Let X = |x|, A = |a|

log(x) - log(a) = log(X/A) + I*(arg(x) - arg(a))
log(X/A) = 1 + log(X*(r+err)/A) ≈ 1 + log1p((X*err - (A+r) + r*(1+X)) / A)

This means we can evaluate f accurately around branch point!

f=0 guaranteed we get the correct branch, which is nice.
There is no need to use complicated W guesses.

lua> a, k = 3+4*I, 5
lua> x = 2*k*pi*I -- guess for W_k(a)
lua> repeat h=f(x)/df(x); x=x-h; print(x, I.abs(h)) until x == x+1e-6*h

(-1.815554248884793+30.714634540472343*I)       1.9462907525577031
(-1.817005891456528+30.71333413897562*I)        0.0019489253984594729
(-1.8170058918466274+30.713334137004896*I)      2.0089622485451688e-009
(-1.8170058918466274+30.713334137004896*I)      0

>>> from mpmath import *
>>> lambertw(3+4j, k=5)
mpc(real='-1.8170058918466274', imag='30.713334137004896')