On Sat, 12 Feb 2011, Rich Hennessy wrote:
> My view of sqrt(x) is a special case of the more
> general definition.? I am not trying to reinvent mathematics.
Then take a serious maths book (not for kids).
You are right, according to Bourbaki, there is no such thing as a
multi-valued function (many good mathematitians strongly dislike
bourbakism, but still...).
Then, what's the domain of sqrt? Unsurprisingly, it's the Riemann surface
of sqrt. It has 2 branch points, 0 and \infty. In a small neighbourhood
of any regular point z, it consists of two sheets; but they are globally
connected in a non-trivial way.
So, there are two distinct points "16" in the domain of sqrt (on two
sheets of its Riemann surface). For one of them, sqrt(16)=4. Let's rotate
it around 0:
z = 16 exp(i*alpha)
When alpha varies from 0 to 2*pi, we arrive to a *different* point - at
the other sheet of the Riemann surface. sqrt(z) becomes -4. Only after the
next resolution (when alpha becomes 4*pi) do we return to the original
point (and sqrt(z) returns to +4).
Maxima works with sqrt's of symbolic expressions. One newer knows what
will be substituted for all variables. When values of variables vary,
analytical continuation can easily transform +4 to -4.
Andrey