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