Re: Composition of series
- To: mathgroup at smc.vnet.net
- Subject: [mg65349] Re: Composition of series
- From: dh <dh at metrohm.ch>
- Date: Tue, 28 Mar 2006 04:05:13 -0500 (EST)
- References: <e08ko1$4c0$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Hi,
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.
Daniel
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 ?
>