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Re: Composition of series

  • To: mathgroup at
  • Subject: [mg65349] Re: Composition of series
  • From: dh <dh at>
  • Date: Tue, 28 Mar 2006 04:05:13 -0500 (EST)
  • References: <e08ko1$4c0$>
  • Sender: owner-wri-mathgroup at

you will agree that a series can only have one expansion point.
Therefore, if we compose seires we may not simply replace the variable 
of one series by a second series. However, if we do a complete 
expansion, everything works just fine: E.g.:

x1 = Series[Exp[x], {x, 0, 3}]
x2 = Series[f[x1], {x, 0, 3}]
gives (InputForm):
SeriesData[x, 0, {f[1], Derivative[1][f][1],
   Derivative[1][f][1]/2 + Derivative[2][f][1]/
     2, Derivative[1][f][1]/6 +
    Derivative[2][f][1]/2 +
    Derivative[3][f][1]/6}, 0, 4, 1]
The expansion point of the second series has been chosen correctly. 
Exp[0]==1. Therefore, f must be expanded around 1.

However, if you do it the wrong way:
x1 = Series[Exp[x], {x, 0, 3}]
x2 = Series[f[xx], {xx, 0, 3}] /. xx -> x1
then you get the error you mentioned.


Geico Caveman wrote:
> I am trying to compose two series with Mathematica :
> One is f1(x) = ...
> The other is x(y) = ...
> ComposeSeries quits complaining that the expansion point of the first series
> is different from the constant term of the second series
> (ComposeSeries::scmp). What I am really trying to do here is to express f
> (x) as a series in Log[x] (x is strictly greater than zero), so the second
> series is simply the series expansion of Exp[y].
> Mathematically, what is the problem with even attempting this operation ?
> Can't I replace x by log(z) in the first series, no matter what ?

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