       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'' x        3
f + f' x + --------- + O[x]
2

then you expect to be able to do operations on this series, e.g.

1/% // Simplify

2
1     f' x    f'    f''    2       3
---- - ------- + (------ - -------) x  + O[x]
f        2         3          2
f      f     2 f

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 + r (-(a f) + f'))

Cheers,
Paul

```

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