Re: [boost] [Review][PQS] Review deadline

From: Geoffrey Irving (irving_at_[hidden])
Date: 2006-06-10 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 unit-correct 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