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difficult/unconventional series expansion

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106259] difficult/unconventional series expansion
  • From: Roman <rschmied at gmail.com>
  • Date: Wed, 6 Jan 2010 06:01:43 -0500 (EST)

Dear group,

I am trying to do an unconventional series expansion, and find it
difficult with Mathematica. Given the function
   S[x_] = Log[E^x-1]/(E^x-1)+(1+1/(E^x-1))Log[1+1/(E^x-1)]
I am looking for the behavior for very large x.
[For the interested: S(x) is the entropy (in units of the Boltzmann
constant) of a harmonic oscillator, with x=\hbar\omega/(kT). So I'm
looking for the low-temperature behavior of the entropy.]

The problem is that simply doing
   Series[S[x],{x,Infinity,1}]
does nothing (since this is not a series expansion in the usual
sense). But I know what I would like to find: the lowest terms of the
"series expansion" are
   (x+1)E^(-x+(2x+1)/(2(x+1))E^(-x))
Do you know how to find this expression automatically in Mathematica?
I am interested in a general technique, not just the results for this
particular function.

Cheers!
Roman.


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