Special delimited oscillators that maxima can't solve it.

On Mon Apr 14 15:11:06 CDT 2008, "Alexey Beshenov" <al at beshenov.ru> wrote:
> On Monday 14 April 2008 21:14, J.C. Pizarro wrote:
>
> > Maxima can't solve the problems of extracting 1000 decimal digits for the
> > following special formulas where its functions are oscillators
> > (and delimited oscillators for defined integrals for one quadrant)
> >
> > * undefined integral and defined integral of below in range below:
> >
> >   1. sin(1/x)         [0+,2/pi]
> >   2. cos(1/x)         [0+,2/pi]
> >
> >   3. sin(1/x^2)       [0+,sqrt(2/pi)]
> >   4. cos(1/x^2)       [0+,sqrt(2/pi)]
> >
> >   5. sin(e^(1/x))     [0+,ln(2/pi)]
> >   6. cos(e^(1/x))     [0+,ln(2/pi)]
> >
> >   7. sin(e^(1/x^2))   [0+,ln(sqrt(2)/pi)]
> >   8. cos(e^(1/x^2))   [0+,ln(sqrt(2)/pi)]
>
> Well, at least sine, cosine, and Fresnel integrals are not hard to compute,
> and implement in Maxima too.
>
> --
> Alexey Beshenov <al at beshenov.ru>
> http://beshenov.ru/

I don't believe that these integrals are not hard to compute and implement
because they are new and were unknown for people until now.

Soon, you will get stuck in somewhere of the implementation in  Maxima
because this new theory is a rare case that difficult you.

The Fresnel integrals are not related to here, in this harder solving.

And i've more updated notes recently:

1. These sines and cosines converge to a constant when go to infinitum.

   They convergence to { 0+, 1-, 0+, 1-, sin(1)+, cos(1)+, sin(1)+, cos(1)+ }
    for the indicated above functions { 1., 2., 3., 4., 5., 6., 7., 8. } when
    go to infinitum.

2. The defined integral of only these sines sin(1/x) and sin(1/x^2) in range
    [0,+inf] is delimited by a constant.

3. Not so in the anoter cases (they are not delimited by a constant).

   There is a new trick:
    You can do them substractions with these convergence's constants pushing
     them to the integrals and so to get simplified new delimited integrals
     thanks to these substractions in integrals.

4. They have some symmetries except the functions 5. and 6.

   About symmetries, they are odd functions {f(-x) = -f(x)} the 1.,
    even functions {f(-x) = f(x)} the 2., 3., 4., 7. and 8., and they have not
    any symmetry relation the 5. and 6.

5. The defined integral of these sines and cosines in range [-inf,+inf] is zero
    for odd functions and 2 * defined integral in [0,+inf] for even functions.

You can test it with plot2d(f(x),[x,0,100]); and plot2d(f(x),[x,-100,100]);

   A good salute, J.C.Pizarro