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MathGroup Archive 1997

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Re: Problems with Limit, Log, E

  • To: mathgroup at smc.vnet.net
  • Subject: [mg9174] Re: [mg9149] Problems with Limit, Log, E
  • From: David Withoff <withoff>
  • Date: Tue, 21 Oct 1997 02:02:47 -0400
  • Sender: owner-wri-mathgroup at wolfram.com

> When I attempt to evaluate the expresssion
> 
>          Limit[ x - Log[1 + E^x],  x -> Infinity]
>
> I simply get back the original expression.  It seems to me that this
> should evaluate to zero.

The built-in Limit function in current versions of Mathematica is
usually only able to compute limits of expressions that have power
series expansions around the limit point.  The expression in this
example has an essential singularity at the limit point:

In[1]:= Series[x - Log[1 + E^x], {x, Infinity, 3}]

Series::esss: Essential singularity encountered in
           1                  4
    Exp[-------- + O[Series`n] ].
        Series`n

                            x
Out[1]= Series[x - Log[1 + E ], {x, Infinity, 3}]

Developing computer algorithms for limits at essential singularities is
a rather difficult problem.  The results of some work on this subject
at Wolfram Research can be found in the Calculus`Limit` package.  This
package is not intended for general use (if it was, it would be loaded
automatically), but it is able to handle the essential singularity in
this example:

In[2]:= << Calculus`Limit`

In[3]:= Limit[ x - Log[1 + E^x],  x -> Infinity]

> For that matter, when I evaluate the expression
>
>         Log[E^x]
>
> I also get back the original expression, instead of x.

This transformation is not done automatically because it is not in
general mathematically correct.  For complex values of x, Log[E^x] can
differ from x by a multiple of 2 Pi I :

In[4]:= Log[E^x] /. x -> 11.3 + 58 Pi I

Out[4]= 11.3

You can force this transformation to be applied anyway using
PowerExpand, which ignores the possible multiple of 2 Pi I :

In[5]:= PowerExpand[Log[E^x]]

Out[5]= x

or you can use ComplexExpand, which is one of the few functions in
Mathematica which assumes by default that x is real:

In[6]:= ComplexExpand[Log[E^x]]

Out[6]= x

> I would appreciate any thoughts concerning these results.
>
> I am running Mathematica 3.0.1 under Windows 95.
>
> Robert Lawrence

Dave Withoff
Wolfram Research


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