On Sat, Feb 12, 2011 at 10:35 AM, Andrey G. Grozin <A.G.Grozin at inp.nsk.su>wrote:
> 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)
>
I thought for sure that I'd no longer add to this learned discussion. I
certainly have little more of value to add. But ...
Yes indeed.
sqrt(z) = sqrt( |z| ) exp(i * alpha/2)
So even in the realm of complex numbers, you can think of it as a single -
valued function if you take the domain to be |z| = [0, inf) and alpha = [0,
4 pi). And it's periodic in 4 pi. As a function of two variables, this
makes good sense. But as a function of one complex variable it may be
bogus because exp(0) and exp(i 2 pi) are the same number, I think. This is
the same as the Riemann sheet idea ... but it looks more analytical. Ok, I
think I'm done for good now. There is a branch of this discussion that has
something to do with maxima, but this ain't it.
-gary
>
> 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
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