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Re: Sqrt of complex number
*To*: mathgroup at smc.vnet.net
*Subject*: [mg126712] Re: Sqrt of complex number
*From*: danl at wolfram.com
*Date*: Fri, 1 Jun 2012 05:20:06 -0400 (EDT)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
*References*: <jpspgr$hdj$1@smc.vnet.net> <201205290948.FAA06757@smc.vnet.net>
On Thursday, May 31, 2012 1:49:23 AM UTC-5, Richard Fateman wrote:
> On 5/30/2012 1:15 AM, Murray Eisenberg wrote:
[...]
> Let's look at the incomplete and incorrect, and see where we (dis)agree.
> >
> > There is nothing incomplete or incorrect (arguably or otherwise) about
> > Mathematica's behavior with Sqrt.
>
> As long as you are talking about Sqrt[] and not "square root", you (or
> Wolfram) is arguably entitled to associate nearly any behavior at all. E.g.
> Sqrt[-2] could return 0 or an error. I believe some version of the
> UNIX library square-root program did this, to avoid complex numbers.
It is not incorrect, as you yourself observe, that Sqrt[] return a single entity rather than a list. Incomplete is a separate issue. I defer to others for the discussion of how various algebraic functions in Mathematica might be applicable to uncovering multivalued "functions" as sets of results.
> If you look up "square root" in wikipedia you get an extensive
> discussion of square roots of numbers, and their principal values.
>
> Unfortunately, the discussion there does not generalize to square roots of
> symbolic items like sqrt(a*x+b) or sqrt(x^2).
I think it does, actually. One simply leaves them alone, as symbolic objects. They can be instantiated by plugging in specific values for the paramete
rs.
This is similar to what you advocate, I believe, for multivalued sets.
> There is a way of thinking about these, and computing with them, for example
> using Root[] expressions.
Mathematica also uses such expressions for roots of transcendental functions. That said, some of the things you might do with, say, quadratic roots, may not generalize so nicely to (infinite sets of) transcendental roots.
> [...]
> What is the definition of the principal root of sqrt((x-y)^2)? how does
> it differ from the principal root of sqrt((y-x)^2)?
One waits until x and y have been supplied with actual values...
> > You might _prefer_ that Mathematica return all possible values of a
> > multi-valued function,
>
> maybe, but maybe not. Maybe there should be another system that does
> multi-valued functions correctly.
Who would build it? Who would use it? What features would it have that existing systems do not? That is to say, what would be the structure and semantics of "multi-valued function objects"? And is this a problem of sufficient merit to warrant the R&D resources needed to address it?
> but that would raise a host of difficulties --
> sure, it is part of the design. That's why certain integrals come out
> wrong because the chosen "principal value" happens to be the wrong value.
> Fixing these errors without doing the mathematics correctly is perhaps
> far more difficult than doing the mathematics right and avoiding the
> errors entirely. This was understood by some of the builders of the
> earlier systems, but Wolfram based his design on the old systems -- with
> their problems -- rather than coming up with a solution.
> [...]
Okay, it was really this that I wanted to comment upon. It is relatively uncommon for an integral to "come out wrong" due to use of principal values. It happens, yes. In some fairly dark corners of indefinite integration. In definite integrals, in cases where path singularities go undetected, or limits at the singularities are not correctly found.
As you surely know, there are indeed researchers who work on ways to avoid principal value issues arising from following paths, e.g. by "unwinding number". If you discuss this with them (as I have), you will learn that they are not particularly closer than the rest of us to addressing these vexing issues of definite integration. I would venture to say that they are, like the rest of us, much closer to retirement than to sorting out such problems.
Daniel Lichtblau
Wolfram Research
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