09/04/2018, 10:31 PM
(This post was last modified: 09/06/2018, 07:10 PM by sheldonison.)
(08/23/2018, 08:35 PM)Chenjesu Wrote: Is there by chance a taylor series for the -1 branch of the lambert w function that doesn't rely on a composite double sum with sterling numbers but looks instead as simply as the 0 branch?
Personally, I rarely use the LambertW function but these links seem to be relevant to your question.
See this math-stack question: https://math.stackexchange.com/questions...ch/2614315
also see this tetration forum thread: https://math.eretrandre.org/tetrationfor...70#pid7470
\( L=\frac{-W(-\ln(b))}{\ln(b)};\;\;\;b^L=L \)
Since I don't personally use the LambertW function series formulation, I was hesitant to answer your question. But one assumes you are interested in this equation for the fixed points. Then you might also want to see this math-stack question. I gave the 2nd answer, which gives a formal series for the approach I prefer to use for the two primary fixed points. I needed both primary fixed points for my fatou.gp program. The xfixed series is an alternative way to get both fixed point given that pari-gp's implementation of the LambertW function is very limited. I use it along with Newton's method to get an answer to any desired accuracy.
\( L_b = \frac{\text{xfixed}\left(\pm\sqrt{-2(\ln(\ln(b))+1)}\right)-\ln(\ln(b))}{\ln(b)} \)
\( f=\sqrt{2(\exp(x)-x-1)};\;\;\;\;\text{xfixed}=f^{-1}(x);\;\;\;f(\text{xfixed}(x))=x; \)
Code:
{xfixed= x
+x^ 2* -1/6
+x^ 3* 1/36
+x^ 4* -1/270
+x^ 5* 1/4320
+x^ 6* 1/17010
+x^ 7* -139/5443200
+x^ 8* 1/204120
+x^ 9* -571/2351462400
+x^10* -281/1515591000
+x^11* 163879/2172751257600
+x^12* -5221/354648294000
+x^13* 5246819/10168475885568000
+x^14* 5459/7447614174000
+x^15* -534703531/1830325659402240000
+x^16* 91207079/1595278956070800000 + ...
- Sheldon