Michel Talon wrote:
> RJF wrote...
>> The approach in the literature is to attack the problem of
>> integrate(R(x,q),x) where R is a rational function, and q satisfies
>> an algebraic equation in x, e.g. q^2= x^4+1. There are several
>> approaches. For one reference, try google {ng elliptic macsyma}
>>
>
> This is not more general that what i advocated, since R(x,q) can be
> written (a(x)+b(x)q)/(c(x)+d(x)q) since q^2 is a polynomial, and then
> multiplying numerator and denominator by (c-dq) you get r1(x)+qr2(x).
>
My assumption was that the more general case included q satisfying an
algebraic equation of higher degree (e.g. q^2 not a polynomial) or the
polynomial P of possibly higher degree. Perhaps some of the cases reduce
to usual elliptic functions, surprisingly. I do not have any first-hand
experience with this, but Ed Ng probably was not dealing with a vacuous
problem.
The fact that stuff is described in references or even texts does not
mean it is described in sufficient form to write a computer program. For
example, consider all the books for freshman calculus that describe
integration. Very few, perhaps none, give an algorithm. For that you
need to look at books on computer algebra, and even then there are gaps.
> Only the second term is non trivial. I reiterate that the general algebraic
> equation P(x,q)=0 is totally useless, since the curve will be hyperelliptic
> or worse, that is the integral cannot be expressed by known special
> functions. The only case of interest boils down to q^2= P(x) with P a
> polynomial of degree 3 or 4. This can always been brought to the form
> P = x^3 + px + q by an homography bringing one of the roots of P at
> infinity, and a translation to kill the x^2 term. This is a canonical form
> for elliptic curves. I don't know what is this stuff about Carlson method,
> what i sketched above is described in all books on Riemann surfaces or
> whatever, for example Springer Introduction to Riemann surfaces, i am
> quite sure it is even in Whittaker and Watson.
>
>
If it is so simple, why has it not been done? Perhaps you would write it?
RJF