Re: Dividing Power Series

From: Gregory Magarshak (contactgreg_at_hotmail.com)
Date: 08/16/04 

Date: 15 Aug 2004 22:17:18 -0700

David C. Ullrich <ullrich@math.okstate.edu> wrote in message news:<9dnvh0hl69n7mr9rbdbq0q34g8p81hrj9q@4ax.com>...
> On 15 Aug 2004 11:12:44 -0700, contactgreg@hotmail.com (Gregory
> Magarshak) wrote:
>
> >Can I divide power series by long division as follows:
>
> yes, at least for z in a disk where both series converge
> and the divisor has no zero.

If the divisor has a zero at the origin, then you get a Laurent series
about 0(for example, cos z / sin z

> pf: the two series define analytic functions f, g in
> this disk. now h = f/g is analytic in this disk, so
> it has a power series. it's easy to show you can
> get the series for f = g*h by multiplying the series
> for g and h formally, and long division is just the
> inverse of formal power-series multiplication [at
> each step the criterion you use to choose a
> coefficient for the quotient is exactly 'make
> the product g*h come out right'.]

That was basically the proof I was alluding to, except I didnt want to
work out the details in "make the product g*h come out right" thing.

-Greg Magarshak

"Computers are useless. They can only give you answers." - Pablo
Picasso.

Relevant Pages

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