       Re: Questions about series and O[x]

• To: mathgroup at smc.vnet.net
• Subject: [mg12797] Re: [mg12782] Questions about series and O[x]
• From: Daniel Lichtblau <danl>
• Date: Fri, 12 Jun 1998 04:05:29 -0400
• References: <199806100704.DAA13695@smc.vnet.net.>
• Sender: owner-wri-mathgroup at wolfram.com

```Matthew D Litwin wrote:
>
> Hi,
>
> I'm working with series with fractional exponents, and have a few
> questions on how to efficiently do certain operations.
>
> 1) Suppose you have a series like f=1 +x^(1/2) +x + x^(3/2) +O[x]^2 What
> is the most efficient way to apply a transformation like  x -> x^(2/3)
> which would yield a series with order O[x]^(4/3). The best I've found
> is to operate on Normal[f] and add the O[x] term by hand, but this
> seems slower than it need be for long series.
>
> 2) Suppose you have two (or three) such series and are taking the
> product, call it P, and you know that the only non-zero terms in P are
> of the form A x^n, where n is an integer. Is there any way to avoid
> computing the fractional terms in P?
>
> Thank you for any suggestions,
>
> -Matt Litwin
> matt at math.ucsb.edu

Not certain, but the answer to 2) is probably "no."

For 1) you might do as follows.

interleave[ll_List, len_Integer] := With[{zeroes=Table[0,{len}]},
Flatten[Map[{#,zeroes}&, ll]]]
replaceWithPower[ser_SeriesData, (pow_Rational|pow_Integer)] := Module[
{g=ser, num=Numerator[pow]},
If [num!=1, g[] = interleave[g[], num-1]];
g[] *= num;
g[] *= num;
g[] *= Denominator[pow];
g
]

For example:

In:= InputForm[f = 1 + x^(1/2) + x + x^(3/2) + O[x]^2]
Out//InputForm= SeriesData[x, 0, {1, 1, 1, 1}, 0, 4, 2]

In:= g = replaceWithPower[f, 2/3]
1/3    2/3           4/3 Out= 1 + x    + x    + x +
O[x]

(* Looks good, but let's check the InputForm to be certain *)

In:= InputForm[g]
Out//InputForm= SeriesData[x, 0, {1, 0, 1, 0, 1, 0, 1}, 0, 8, 6]

To see why this works:

In:= ??SeriesData
SeriesData[x, x0, {a0, a1, ... }, nmin, nmax, den] represents a power
series
in the variable x about the point x0. The ai are the coefficients in
the
power series. The powers of (x-x0) that appear are nmin/den,
(nmin+1)/den,
... , nmax/den.