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From: Geoffrey Irving (irving_at_[hidden])
Date: 20060610 13:50:24
On Sat, Jun 10, 2006 at 09:54:14AM +0200, Gerhard Wesp wrote:
> On Fri, Jun 09, 2006 at 06:19:58PM 0700, Geoffrey Irving wrote:
> > polynomial regression). All the nice Taylor series example seem unitless.
>
> They have to be, don't they? Because you add up different powers of the
> argument. I took this once as a "heuristic" explanation to myself why
> the transcendental functions only work for dimensionless arguments.
>
> On the other hand, the square root can be approximated by a series as
> well, and this function does make sense with dimensional arguments.
That's cool. Square root isn't an example either because it has a branch cut
and therefore doesn't have any infinitely converging series. If you want to
use a series square root, you need to remove the units first.
In general, if f(z) is a total analytic unitcorrect function, then we have
f(z a) = f(z) a^p
for some p. If p is fractional or <0, f is not total, so p is a positive
integer. But then f(inf) = f(z inf) = f(z) inf^p = inf, so f is a polynomial.
That makes your heuristic argument rigorous: the only examples are polynomials.
Geoffrey
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