Billinghurst, David (RTATECH) wrote:
>> From: Dieter Kaiser
>
>
> Great. I am happy to do some detailed coding and testing for
> special functions. I was looking at your new code last night
> - trying to work out how to integrate Jacobi elliptic functions.
> I didn't find a neat, general solution but it looked feasible.
>
> I was also wondering how to add a hook/property to integrate
> (simple) expressions containing a special function, such as:
> * integrate( sin(x)/x, x)
> * integrate( x^(n+1).bessel_j(n,x), x)
> * integrate( jacobi_sn(u,m)^n, dx)
>
Well, speaking of elliptic functions, i think a worthy addition would be
recognize the fact that any integral of the form
integrate(R(y(x),x),x)
where R(y,x) is a rational function of x and y, and y(x) is the square root
of a polynomial in x of degree 3 or 4 without double roots is elliptic.
The traditional case is
integrate(1/sqrt((1-x^2)*(1-k^2*x^2)),x)
but the general case boils down to the same after systematic reductions. Of
course in case of a double root the integral becomes trigonometric. If the
relation between x and y is more complicated (meaning the curve R(y,x)=0 is
not elliptic) then it is certain that the integral cannot be computed on
elementary or elliptic functions.
--
Michel Talon