On Fri, Feb 11, 2011 at 6:41 PM, Michel Talon <talon at lpthe.jussieu.fr>wrote:
> Rich Hennessy wrote:
>
> > I don?t know why I jump in to say anything on this, but a function can
> > have
> > only one value. If it has more than one it is not a function, by
> > definition
> > of mathematical functions. So sqrt(16)=4, not 4 and ?4 too.
> >
> > So sqrt(-16) = 4*%i
> >
> > I take the positive imaginary axis as the preferred answer for completely
> > arbitrary reasons. You know what positive imaginary means right? The
> > part
> > of the imaginary axis ABOVE the real axis. I say it thus because I like
> > it better.
> >
>
>
> Rich, i must say i agree completely with prof Fateman discussion on this
> point and thus disagree with you. The notion of a function having just one
> value on a given domain to a given range is certainly the notion commonly
> used in elementary courses, but so-called multivalued functions have been
> used continuously by most great mathematicians since at least 19^th century
> up to now, particularly in algebraic geometry. For example sqrt(-16) is
> 4*%i if you continue analytically from sqrt(16)=4 through the upper half
> plane, but -4*%i if you continue through the lower half plane. It jumps
> abruptly on the (x<0) axis. Since in a CAS when you consider a square root,
> it is generally the square root of some complicated expression of x it is
> basically impossible to enforce the rules that students learn in elementary
> courses. To reconcile single valuedness and the obvious multivaluedness of
> the sqrt, one considers "Riemann surfaces" where basically you duplicate
> the x plane into two planes touching at 0 and infinity. So a point in this
> surface is a couple (x ,sqrt(x)), and (x, -sqrt(x)) is another point.
> Now you can define a single valued sqrt on *that surface* which is simply
> the projection (x,sqrt(x)) -> sqrt(x). This is a bona fide function in the
> Bourbaki sense, or in the elementary school sense. The other projection
> (x,sqrt(x)) -> x is called a branched covering of the complex plane.
> Whether
> this sort of ratiocination is able to clarify your objection, i don't know,
> but basically Fateman's argument that a degree N equation has N roots is
> all that is really to understand. By analytic continuation in parameters of
> the polynomial these roots exchange (this is related to Galois theory) so
> there is no way for a CAS to really distinguish them. In particular, as
> Fateman says, this rule of elementary school sqrt(x^2)=abs(x) which is true
> for x real, is false when you go to complex x. I would be much happier
> with sqrt(x^2)=x which is always true, provided you understand that sqrt
> has
> two values and you have expressed one of them.
>
>
> --
> Michel Talon
>
I think we can all accept that.
The semantic question comes up in a practical discussion:
What do you expect your CAS system to do?
sqrt(16) = 4 or {4, -4} ?
sqrt(-16) = #error# or +4%i or -4% ?
... and I admit that I lost sight of the question of symbolic manipulations,
at which point I stopped thinking about it.
I'm sure this is all worked out in some consistent way in maxima. One
needs to RTFM or ask on the forum, and accept the answer.
Don't we have better things to discuss, like revolution in Egypt or
basketball losing streaks? Bye for now!
-gary