Michel Talon wrote:
> Hello,
>
> i have found a nice online article which explains in great detail the
> algorithm from Legendre allowing to reduce general elliptic integrals
> to standard elliptic integrals of first, second and third form.
>
Unfortunately it doesn't. The "steps" indicated are not explained
algorithmically, and in fact there may be
extraordinary difficulty in making them algorithmic. That is why the
programs to do them have probably not
been written in the last several decades. As a very simple example, I
quote from the document,
"The first integral is simply the integral of a rational function, which
anyone who learned undergraduate integral calculus
<http://everything2.com/title/integral%2520calculus>; should be able to
integrate."
In fact, undergraduate integral calculus teaches students to integrate
rational functions whose denominators are either linear, quadratic, or
factor into
linear and quadratic factors. In principle one can find such factors
but may require finding representations for (all) the roots of a
polynomial of
degree n, and simplifications relating to combinations of them. You may
find this "simple" as does the author of this document, but then how do
you explain the several Ph.D thesis written on this topic ... explicit
integration of rational functions?
[Robert G. Tobey; Michael Rothstein, Barry Trager]
as well as numerous papers by others.
The other steps may be easier or harder to automate, but the author
probably is not used to thinking algorithmically.
> http://everything2.com/e2node/elliptic%2520integral%2520standard%2520forms
> Here the aim is to arrange so that branch cuts are symmetric with respect to
> 0, and this is achieved through clever tricks.
>
By "clever tricks" I suppose you mean "I don't know how this can be done
in general". That does not mean it
is easy or hard, of course. We have just pointed out that if a
mathematician says something is done "simply"
is sometimes wrong. What about the other side of the coin, if something
is tricky or hard?
Consider asking a mathematician to (say) prove that a polynomial of
substantial degree is or is not irreducible over the rationals.
This is easy if the mathematician knows that it can be done by a
computer program and has access to such a program.
In the absence of such knowledge, is the mathematician allowed to claim
that it is a hard problem, or can be done by tricks?
The point I'm trying to make is, a constructive algorithmic solution to
a problem is quite different from what is typically provided in a
mathematical "explanation". In some cases, mathematical "methods"
include as sub-components, steps which are not truly understood
algorithmically and have not ever been programmed up to this time.
Sometimes a step is even provably not computable, though whether the
mathematician even understands the theory of computability is not to be
taken for granted.
I know of no freshman calculus textbook that has an algorithm for doing
antidifferentiation. Even in the case of rational functions.
But I have not made an exhaustive search either.