HP49-50G,VER16.2 Lambert Funktion Wk(x), k=k= 0, ±1, ±2, ±3, ±4..., x real or complex

[Gil]

(01-14-2024 05:03 PM)Albert Chan Wrote:  

(01-14-2024 02:50 PM)Gil Wrote:  Yes, it's the algorithm that I found with the given reference.

If you meant given reference here, it is guaranteed to return correct branch.

x = W(a, k), we rewrite definition of branch k as function.

f(x) = x + log(x) - log(a) - 2*k*pi*I
f'(x) = 1 + 1/x

Newton: x ← x - f/f'

Solve for f(x)=0 thus guaranteed correct branch.

Without signed zero, code assumed real a == a+0*I
For illustration, here is lambertw code that support signed zeros.

lua> require'expW'
lua> I.W(-0.1, -1)
(-3.577152063957297-0*I)
lua> I.W(I.conj(-0.1), -1)
(-4.44909817870089-7.3070607892176085*I)

Technically, W_-1(-0.1) does not exist. (we have 2 different limits!)
By convention, we returned W_-1(-0.1+0*I), which does exist.

For comparison, W₀(-0.1) does exist.

lua> I.W(-0.1, 0)
(-0.11183255915896298+0*I)
lua> I.W(I.conj(-0.1), 0)
(-0.11183255915896298-0*I)

Note that 2nd result is conjugate of 1st.
In general: W(a,k) = conj(W(conj(a),-k))