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MathGroup Archive 1994

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Re: Series involving logarithms

  • To: mathgroup at yoda.physics.unc.edu
  • Subject: Re: Series involving logarithms
  • From: withoff (David Withoff)
  • Date: Mon, 9 May 1994 13:28:44 -0500

> 	Dear MathGroup,
> Mathematica 2.2 apparently support the series involving logarithms:
> 
> Mathematica 2.2 for DEC RISC
> Copyright 1988-93 Wolfram Research, Inc.
>  -- Terminal graphics initialized -- 
> 
> In[1]:= Series[x^x,{x,0,4}]
> 
>                              2  2         3  3         4  4
>                        Log[x]  x    Log[x]  x    Log[x]  x        5
> Out[1]= 1 + Log[x] x + ---------- + ---------- + ---------- + O[x]
>                            2            6            24
> 
> 
> I would expect that Logs be treated as constants when they appear in
> a series. So far I would expect that Log[1+x Log[x] + O[x]^2] = 
> x Log[x] + O[x]^2, but
> 
> 
> In[2]:= Log[1 + x Log[x] + O[x]^2]
> 
>                                                                            2
> Series::lss: Logarithmic singularity encountered in Log[1 + Log[x] x + O[x] ].
> 
>                                2
> Out[2]= Log[1 + Log[x] x + O[x] ]
> 
> It is strange because Log is analytic near 1, it is as good as Exp
> near 0, yet for Exp there is no problem:
> 
> 
> In[2]:= Exp[x Log[x] + O[x]^2]
> 
>                            2
> Out[2]= 1 + Log[x] x + O[x]
> 
> 
> Does anyone know how the Series::lss message is generated and is there
> any way to teach mathematica to deal with logarithms properly rather
> then substitute them by constants in the intermediate calculations?
> 
> Alexander Belopolsky.

======================================================================

The Series::lss message is generated when Mathematica attempts to
compute the logarithm of a series in which the coefficients depend
on the expansion variable.  A power series expansion in which the
coefficients depend on the expansion variable is already a
bit peculiar, since it means that the expansion isn't a power
series expansion at all, but something else entirely.  Such
expansions are convenient in many situations, which is the reason
they are not immediately rejected, but some manipulations, such
as the one you mentioned, will not always give useful results.
The solution to this problem is to add more general types of
series expansions other than power series expansions.  Doing
this correctly is a formidable undertaking, and will probably
have to wait until a future release of Mathematica.

Dave Withoff
Research and Development
Wolfram Research





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