On Fri, Jul 15, 2016 at 6:23 PM, Bill Gosper <billgosper@gmail.com> wrote:
OUCH! I can't believe my senility. Change the subject to the Limits of my credibility. I thought I was typing Limit[ArcSinh[x Log[x] - x] - Log[Log@x]^1 - 1/E^E^E^E^E^x, x -> \[Infinity]] and that no matter how slightly I perturbed my question, the answer swung from -∞ to ∞. Rob Corless, of LambertW fame, was able to expand the LHS after loading Maple's MultiSeries. I'll send his interesting result
Out[488]= ArcSinh[x Log@x - x] - Log[Log[x]] == Log[2] + Log[x] - Log[Log[x]/(-1 + Log[x])] + 5/(96 x^6 (-1 + Log[x])^6) - 3/(32 x^4 (-1 + Log[x])^4) + 1/(4 x^2 (-1 + Log[x])^2) + ... which is peculiar for its expansion variable x ln x - x, and for not expanding ln(ln x/(ln x -1)). But with this clue we can coax the answer out of Mathematica by substituting ln x -> L+1: ArcSinh[x Log[x]] - x - Log[Log[x]] == ArcSinh[L x] - Log[1 + L] and then pretending L and x are independent variables (modulo ln 1/x -> - L - 1): Normal@Series[%, {x, \[Infinity], 6}, {L, \[Infinity], 6}] Out[540]= 1 + 1/(6 L^6) - 1/(5 L^5) + 1/(4 L^4) - 1/(3 L^3) + 1/(2 L^2) - 1/L + L + 5/(96 L^6 x^6) - 3/(32 L^4 x^4) + 1/(4 L^2 x^2) + Log[2] which is about what nhayat would have done, but legibly. along a minor surprise,
This is a job for LambertW! In[489]:= Solve[x (Log@x - 1) == y, x] Out[489]= {{x -> y/ProductLog[y/E]}} So ArcSinh[x Log@x - x] - Log[Log[x]] - Log[x] becomes Out[499]= -1 + ArcSinh[y] - Log[1 + ProductLog[y/E]] - ProductLog[y/E] Unfortunately, Series completely blows it from here, so messily that it crashed my kernel when I tried to Simplify. But, surprisingly, In[2]:= Limit[-1 + ArcSinh[y] - Log[1 + ProductLog[y/E]] - ProductLog[y/E], y -> \[Infinity]] Out[2]= Log[2] (Note involuntary 499->2 !) Limit is flaky, but not nearly as bad as Series. Does it have its own secret Series? --rwg I thank Julian for independently solving this problem and reaching similar conclusions. and to WRI Support a raft of pathology.
--rwg It looks like MultiSeries has most of the functionality of nhayat, but with some peculiarity.
On Sun, Jul 10, 2016 at 5:28 PM, Bill Gosper <billgosper@gmail.com> wrote:
In[152]:= Limit[ArcSinh[x Log[x] - x] - Log[Log@x]^1, x -> \[Infinity]]
Out[152]= -\[Infinity]
In[167]:= Limit[ArcSinh[x Log[x] - x] - Log[Log@x]^(1 - 1/E^E^E^E^E^x), x -> \[Infinity]]
Out[167]= \[Infinity]
There was a brief shining moment when developmental Macsyma could actually do these, using Bill Dubuque's nhayat asymptotic expansion system, inspired by a certain moderator. --rwg