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