On 2/20/2011 11:29 AM, Michel Talon wrote:
> Dan Stanger wrote:
>
>> Integrals like this can often be done more easily with the Euler
>> substitution, here is a link: http://eom.springer.de/e/e036590.htm .
> I have looked at the springer link. So this substitution boils down to the
> fact that every degree 2 curve is of genus 0 and thus has rational
> parametrization. It is another instance of the fact that any function
> rational in sin and cos can be reduced to purely rational by using
> the tangent of the half angle, as we discussed recently.
>
> There also exists an Euler trick for genus 1 curves (cubics) which
> has to do with the natural addition on the cubic, and also with square roots
> but i don't remember exactly what it is. There is a big discussion of this
> trick in a paper by van Moerbeke and Horozov called "The full Geometry of
> Kowalewski's top and (1, 2)-abelian surfaces".
>
Joel Moses' 1967 MIT Phd thesis (LCS-TR-047) has, on page 84, a
description of
one of the methods in his stage 2,
"Notes - In the case where the integrand is a rational function of
trigonometric
functions of x all the problems can be reduced to rational functions.
The choice
of the transformation [ several are proposed, but transformation VI is the
tan of half angle] governs the simplicity of the resulting rational function
and the final answer. The transformation in Step VI above which is always
applicable in these situations frequently leads to a great deal of work and
to complex results. Fortunately, a number of simpler transformations
such as
those of steps III, IV, and V are easily recognized and are usually
applicable.
...
SAINT included all of the transformations given above. ..."
Moses' thesis work was included in Macsyma and hence Maxima.
I don't know if this method is still used.
RJF
reference:
http://publications.csail.mit.edu/lcs/pubs/pdf/MIT-LCS-TR-047.pdf