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Re: Re: Bug in interpretation of mma Series[] command?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg2326] Re: [mg2268] Re: Bug in interpretation of mma Series[] command?
*From*: Paul Abbott <paul at earwax.pd.uwa.edu.au>
*Date*: Tue, 24 Oct 1995 02:14:50 -0400
*Organization*: Dept of Physics, University of WA
Richard Mercer <richard at seuss.math.wright.edu> wrote:
>From the viewpoint of most users, I think the ideal situation would be
>for
>
>Exp[-a y^2] * Series[1/(1 + y^2), {y,0,5}]
>
>to act like
>
>
>Exp[-a y^2] * Normal[Series[1/(1 + y^2), {y,0,5}]]
>
>Reasons:
>(1) Most average (nonexpert) users do not have a mental model
>corresponding to a SeriesData object; they think of the result of a
>Series command as being a polynomial and expect it to behave that way
>in calculations.
When you compute a series, say,
f[x] + O[x]^3
2
f''[0] x 3
f[0] + f'[0] x + --------- + O[x]
2
then you expect to be able to do operations on this series, e.g.
1/% // Simplify
2
1 f'[0] x f'[0] f''[0] 2 3
---- - ------- + (------ - -------) x + O[x]
f[0] 2 3 2
f[0] f[0] 2 f[0]
If you think of the result of a Series command as being a polynomial you
lose this functionality.
Also, the fact that the syntax f[x] + O[x]^3 coerces Taylor series
expansion of f[x] is elegant and useful. It also means that if you
write:
Exp[-a y^2] 1/(1 + y^2) + O[y]^6
you get
2
2 a 4 6
1 + (-1 - a) y + (1 + a + --) y + O[y]
2
>(2) The structure Series[Exp[-a y^2] * 1/(1 + y^2), {y,0,5}] is
>available and much more natural if you want the exponential converted
>to a series.
So is the syntax
Exp[-a y^2] Normal[1/(1 + y^2) + O[y]^6]
>The best solution would seem to be a user-settable "switch" that would
>apply Normal to the output of all Series commands. This would
>presumably satisfy all those who expect Series objects to act like
>polynomials in this and other situations.
You do have a user-settable "switch". It is Normal and you can use this
to turn a Series into a polynomial whenever you need this behaviour.
Here is an example that demonstrates what a nice design feature this
syntax is. Suppose that you want to factor off some (exponential)
asymptotic behaviour. You can use the syntax:
Exp[a r] Normal[Exp[-a r] f[r] + O[r]^2]
a r
E (f[0] + r (-(a f[0]) + f'[0]))
Cheers,
Paul
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