Mistake about ProductLog expansion at Infinity
- To: mathgroup at smc.vnet.net
- Subject: [mg44875] Mistake about ProductLog expansion at Infinity
- From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
- Date: Thu, 4 Dec 2003 03:04:39 -0500 (EST)
- Sender: owner-wri-mathgroup at wolfram.com
Concerning the series expansion of ProductLog at Infinity, Mathematica
makes an incorrect statement (and also does something very curious in the
process):
In[1]:= Series[ProductLog[x], {x, Infinity, 0}]
Out[1]=
<<stuff which is correctly based on the expansion at Infinity>> + O(1/x)^1
The stuff above (if simplified assuming that x > 0) shows all the terms in
the series up through the term having denominator Log[x]^6. So you ask
"Then what's wrong with Out[1]?" Answer: The claim that what remains after
<<stuff>> is O(1/x)^1. Now it certainly is true that, after <<stuff>>, the
remainder -> 0 as x -> Infinity. But O(1/x)^1 says something much more
specific than that. And that's the problem. The remainder -> 0 far more
slowly than 1/x. Indeed, x(remainder) -> -Infinity as x -> Infinity. So why
did Mathematica make this mistake?
Now for the curious part:
For Series[ProductLog[x], {x, Infinity, 0}], if Mathematica had just given
Log[x] - Log[Log[x]] + O(1/x)^1, I wouldn't have been so surprised. Of
course, the result is wrong, just as noted above, but I could at least have
guessed how Mathematica might have made this specific mistake. So the
curiosity, IMO, is why Mathematica felt compelled to give the terms up
through that having specifically denominator Log[x]^6, but then decided
that no other terms were needed! Why on earth would Mathematica think that?
David Cantrell
P.S. For reference, the Wolfram Functions site gives the correct full
expansion for ProductLog[z]:
Log[z] - Log[Log[z]] - Sum[((-1)^ k/Log[z]^k) Sum[(StirlingS1[k, k - l +
1]/l!) Log[Log[z]]^l, {l, 1, k}], {k, 0, Infinity}]