[math-fun] The Limit of my credulity
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
Let us (and wolfram.com) into the secret --- what does Log@x mean? https://reference.wolfram.com/language/tutorial/Operators.html WFL On 7/11/16, 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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I'm going to guess that it means Log iterated x times, with some interpretation when x is noninteger. -- Gene From: Fred Lunnon <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Sunday, July 10, 2016 5:48 PM Subject: Re: [math-fun] The Limit of my credulity Let us (and wolfram.com) into the secret --- what does Log@x mean? https://reference.wolfram.com/language/tutorial/Operators.html WFL On 7/11/16, 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
I think it's Mathematica's prefix notation for Log[x] https://reference.wolfram.com/language/ref/Prefix.html Tom Fred Lunnon writes:
Let us (and wolfram.com) into the secret --- what does Log@x mean?
https://reference.wolfram.com/language/tutorial/Operators.html
WFL
On 7/11/16, 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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 2016-07-10 20:55, Tom Karzes wrote:
I think it's Mathematica's prefix notation for Log[x]
https://reference.wolfram.com/language/ref/Prefix.html
Tom
Yeah, sorry, just being lazy on input, and then forgetting to clean up for general consumption. Note that the prefix form does more than save one character. It often saves you from spacing over whole expressions to where to insert the matching ]. The downside is you need to know operator precedences, or wind up needing extra parens. --rwg Meanwhile, is %152 right? And how many E^E^ do we need to reverse that sign?
Fred Lunnon writes:
Let us (and wolfram.com) into the secret --- what does Log@x mean?
https://reference.wolfram.com/language/tutorial/Operators.html
WFL
On 7/11/16, 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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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 along a minor surprise, 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
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
participants (5)
-
Bill Gosper -
Eugene Salamin -
Fred Lunnon -
rwg -
Tom Karzes