[GiNaC-devel] Powers of exponents

[Francois Maltey]

Hello everybody,

I'm a new ginac user because I discover sage. So you might excuse my 
point of view if my arguments are out of ginac purpose. And this message 
isn't direct reponse for this patch.

// 1 //

I understand that ginac operates over complex analysis as the other 
computer algebra systems.

In this case it's very curious to write exp(u)^v == exp(u*v).
Fine choices of "branch cuts" may allow this point of view for a local 
study,
but it isn't usual mathematics for general purpose.

You understand I don't like the exp(u)^v == exp(u*v)

// 2 //

It' bad if exp(x)/exp(x) remains : sage reduces sin(x)/sin(x)  == 1 as 
usual.
I don't know the inner algorithms of ginac but I suppose that all 
function calls as sin(x) are seen as a new variable in the expression 
which is a fraction with a lot of variables, even if someones as sin(x) 
and cos(x) are linked together by cos(x)^2+sin(x)^2==1.

So exp(x)/exp(x) must be simplified in 1.

// 3 //

Look at sin(x) and cos(x).

Sometimes the user prefers the expanded formula with Tchebytchev polynomials
(cos(2x)+1)/(sin(2x)) == (2 cos(x)^2)/(2 sin(x) cos(x)) == cos(x)/sin(x) 
== cotan(x).

Othertimes the user wants to combine 2 cos(x)^2 into the (almost) linear 
form cos(2x)+1.
Computer algebra systems don't have any automatic transform
but the user calls the expand or the combine function for theses 
opposite purposes.

Calculus are similar with exp : both transforms exp(x)^2 <==> exp(2*x) 
are useful.

I observe that ginac respects algebraic user input :
by example there are very view transform with

x = 2*t/(1-t^2) ;
x.subs(t=x) == 4*t / (1 - (2*t/(1-t^2))^2)

Then the user calls expand, "simplify_fractions" or others functions if 
he wants an other form of this input.

power and exp functions might be in the same case :
exp(x)^2 remains exp(x)^2 and exp(2*x) remains exp(2*x).
Then an expand call translates exp(2*x) to exp(x)^2
and a combine (or an other name) translates back exp(x)^2 to exp(2*x)

If it does so, ginac respects also the user choice for the exp function 
as it does for the fractions.