R Fateman wrote:
> While I agree that Maxima should not ask the sign of an expression when
> it can be figured out,
> the problem of ridding the system of these questions when the sign of an
> expression is NOT apparent
> cannot be solved by simply forbidding it, or blaming someone.
>
> The earliest and simplest version of this was probably introduced by
> indefinite integration procedures where
> the choice was between expressing the answer as an arctangent or log.
> Which to use affects the form of
> the answer (do you want to avoid complex constants?) as well as some
> other properties, like continuity [I think].
Maple has a conversion procedure:
> convert(arctan(x),ln);
1/2 I (ln(1 - x I) - ln(1 + x I))
>
> So if you want to eliminate asksign, you can do so by adding gratuitous
> discontinuities, or find something cleverer all around.
I do not agree with that. Discontinuities are precisely introduced by
arbitrary definitions like the little children one:
"
sqrt(x) is "the positive square root of x" for x real >0 and then the
analytic continuation of that in the complex plane cut on x real and x<0.
"
Then sqrt() is obviously discontinuous on the negative real axis. But
the reality is that sqrt() is defined on the "Riemann surface" of the square
root and is perfectly continuous and even analytic here. The only
peculiarities of this "surface" are branch points at x=0 and x=infinity,
the "branch cut" joining these two points is completely arbitrary, and
the "sign" consideration totally irrelevant. For example, if you consider
sqrt(x-%i) what will be the relevance of sign considerations for (x-%i)?
Even worse, if you consider cube roots instead of square roots, then each
cube root has 3 values differing by j and j^2, and so on. To summarize, i
don't think it is the job of a computer algebra system to choose branches
and such stuff, it is to do algebra, that is to do formal computations.
In particular, not wanting to expand sqrt(xy)=sqrt(x)sqrt(y) on the pretext
that the signs on right and left may differ is negating the benefits of a
computer algebra system.
Maple has the option to do such things, for example simplify(expr, symbolic)
"
The symbolic option indicates that formal symbolic manipulation of
expressions is allowed without regard to the analytical issue of branches
for multi-valued functions. For example, the expression sqrt(x^2) will
simplify to x under the symbolic option whereas without this option the
simplified result must take into account the different possible values of
the (complex) sign of x.
"
To say the truth, this features works in a very limited way in Maple, i.e.
it doesn't see very obvious simplifications. I have enjoyed better luck with
Maxima in several cases.
--
Michel Talon