(07/19/2022, 06:51 AM)bo198214 Wrote: I thought you just would calculate the base from the fixed point via \(b=z^{\frac{1}{z}}\) and then apply \(\text{LambertW}(-\log(b))/(-\log(b))\) with the right branch?
When I made the animated picture of the primary fixed points, I used these branches \(k\) of the LambertW:
For the attracting fixed point \(z\) in the upper half plane excluding the real line (left of \(e\)) use \(k=1\). Otherwise use \(k=-1\).
For example for \(z_1=2\) is \(b=\sqrt{2}\) then \(z_2 = \text{LambertW}_{-1}(-\log(b))/(-\log(b)) = 4\)
Oh my God I totally missed your comment BO!
YES THAT IS WHAT I WANT!
I want that in PARI-GP, and everything I've read says that pari doesn't support complex Lambert W, and additionally only has the principal branch.
Thank you though, I want \(W_{-1}\) not \(W_1\), I knew I was screwing something up.
You answered my question, but I'm still confused how to run this in Pari-GP. That's still confusing me.
But yes, that's what I want! Thank you, that's the mathematical problem explained.
How do I write
Code:
LambertW(sqrt(2) + O(y),-1) = 4 + O(y)\\Where we can gather taylor terms in the same manner any transcendental function works in pari. (I'm kind of a stickler for this--I don't want a numerical value, I want a polynomial value).
So I want, LambertW(sqrt(2) + x,-1) to be a polynomial expansion about the point \(\sqrt{2}\) in the variable x. If I can get an efficient way of doing that, that isn't my hacked together way now. Omg, I wouldn't know how to say thank you.
Tell me how to program LambertW(.,-1); ffs. Lambert has always been a handicap of mine.
