(28/48/50) Lambert W Function

[Albert Chan]

(04-03-2023 10:47 PM)Albert Chan Wrote:  cos(θ) = ln(|a|/r)/r
sin(θ) = (T-θ)/r

We assume k positive --> both r and x imag part positive too.

We can setup solver to get r, then W_k(a)
Let c = cos(θ), s = sin(θ)

s = (T-θ)/r
θ = T-r*s = T - r*√(1-c²)

s ≤ 1      → min(r) = T-θ > T-pi

lua> g = fn'r,c: c=log(A/r)/r; acos(c) - (T-r*sqrt(1-c*c))'

lua> S = require'solver'
lua> r = S.secant(g, T-pi, T+pi, nil, true)
29.20162910030975
35.48481440748934
30.766823278237727
30.76703434746101
30.767034374835603
30.767034374835603
lua> c = ln(A/r)/r
lua> r * I.new(c, sqrt(1-c*c)) -- = W₅(3+4*I)
(-1.8170058918466274+30.713334137004896*I)

I setup function comparing angles. For |k| ≥ 2, r only have 1 real root.