lambertw, all branches

[Albert Chan]

Quote:With signed-zero, arg(z) + arg(conj(z)) = 0

Without signed-zero, above might not hold.
If a is negative real, conj(a) returns itself, we get sum = pi + pi ≠0

In this case, this does not hold either: W_k(a) = conj(W_-k(conj(a)))

One way to get sum=0 is to to use ±eps for ±0 (★)

p2> z = mpc("-1E-330", "1E-9999")

p2> W(z, -1)
(-766.494908744112 - 1.00130634441663e-9669j)
p2> conj(W(conj(z), 1))
(-766.494908744112 - 1.00130634441663e-9669j)

p2> W(z, 1)
(-766.494942472642 + 6.2913931263013j)
p2> conj(W(conj(z), -1))
(-766.494942472642 + 6.2913931263013j)


(★) with eps hack, we may get into another ugly situation.

What if eps underflowed to zero? Then, the sign is lost!

Luckily, mpmath (by design) will not overflow/underflow.

https://mpmath.org/doc/current/technical.html Wrote:Mpmath uses arbitrary precision integers for both the mantissa and the exponent,
so numbers can be as large in magnitude as permitted by the computer’s memory.

Still, I think better solution is to properly add signed-zero to mpmath.

fredrik-johansson Wrote:Should negative zero (-0) be supported? It is virtually supported already; the object 'fnzero' is defined in libmpf.py (though not used anywhere). It just needs to be supported in standard operations, and elsewhere "x ==fzero" needs to be replaced with something else so that -0 is not forwarded to places where it shouldn't go ...