[Maxima-discuss] Canonical exponential forms

[Mike V. <mic...@gm...>]

Hello all,

I was wondering a few things in general and their applications to Maxima.
Disclaimer, I'm still using Maxima 5.36.0, for reasons I'm not so sure
myself.

In general, is it canonical (providing a 1 to 1 mapping of expressions to
their meanings) to write all exponentials as:

base^(exponent expression) --> exp(log(base) * (exponent expression))?

Converting back ("simplifying") can be ambiguous:
exp(log(a) * log(b)) --> a^log(b) or b^log(a)?

There are times when I want the "canonical form", but I am not certain if
it is truly a canonical form. If it is, should the procedure to convert it
be as simple as exponentiating the log of the base multiplied by the
original exponent? I want to be careful is all. This seems to work on all
branches; consider a base of negative 1.

(-1)^x --> exp( log(-1) * x) --> exp( %i * %pi * (1 + 2*n) * x )
where n is an arbitrary integer

I think converting to this form will in general cause all multiplication of
power/exponentials to collapse into a single exp(sum of individual
log(bases)). I am trying to improve the automatic simplification of a
product of powers:

a^x * b^x --> exp((log(a)+log(b))*x) --> ONLY IF x is real? -->
exp(log(a*b)*x)
A quick numerical check shows:
(%i1) float(rectform( exp((log(-1) + log(-2)) * x))), x=%i;
(%i2) float(rectform( exp((log(-1 * -2)) * x))), x=%i;
(%o1) 0.001193223591294758*%i+0.001436509595299189
(%o2) 0.6389612763136348*%i  +0.7692389013639721
I get a similar discrepancy using x=2*%i*%pi. So a real argument appears at
least necessary.

In Maxima 5.36.0:
(%i3) radcan((-1)^x * (-a)^x);
(%o3) (-a)^x * (-1)^x;
(%i4) declare(x, real)$
(%i5) radcan((-1)^x * (-a)^x);
(%o5) (-a)^x * (-1)^x;
(%i6) is(equal(radcan(exp((log(-1) + log(-2))*x) - exp(log(-1*-2)*x)),0));
(%o6) unknown;
(%i7) load(to_poly_solve)$
(%i8) %solve(exp((log(-1) + log(-2))*x) - exp(log(-1*-2)*x) = 0, x);
(%o8) %union([x=%c5])

Something here feels wrong. %Solve says any complex number works. Rectform
shows different results. I think I trust %solve's answer more than the
float(rectform()) answer since %solve tends to work with all branches and I
think the call to float is causing something stupid to happen. Possibly I
misused is(equal()).

Anyway, I just wanted to know what the community thinks of yet another
canonical form, if it is even that. Would this even be a valuable addition
to Maxima?