Special delimited oscillators that maxima can't solve it.

On 2008/4/16, Richard Fateman <fateman at cs.berkeley.edu> wrote:
>  I haven't followed all of this, but by a change of variables,
>  integrate(cos(1/x),x,0,2/%pi)
>  is integrate(cos(t)/t^2,t, %pi/2, inf);
>
>  this latter can presumably be integrated numerically by filon type formula
>  to any desired accuracy.
>  See
>  http://www.cs.berkeley.edu/~fateman/papers/fun-filon.pdf
>  for a start on this, and references, and maxima code..
>
>  RJF

You did a mistake, your change of variables t=1/x doesn't match symbolically
 your integrals. I can't presume anything "mathematically" with human
 suppositions because otherwise it can be incorrect simplification.

  integrate(cos(1/x),x,0,2/%pi) is NOT integrate(cos(t)/t^2,t, %pi/2, inf);

The correct simplification by the substitution method
  integrate(f(x),x) = integrate(f(g(t))*g'(t),t) with x = g(t) = 1/t ,
   g'(t)=-1/(t^2) and f = "cos(1/" is the below

      integrate(cos(1/x),x) = integrate(cos(1/(1/t))*-1/(t^2),t)

      integrate(cos(1/x),x) = - integrate(cos(t)/(t^2),t)

  So, you mistaked your matching by the sign symbol '-'.

   integrate(cos(1/x),x,0,2/%pi) = - integrate(cos(t)/(t^2),t,%pi/2, inf)

------------------------------------------------------------------------------

My previous handwritten simplifications by parts's method of
  integrate(sin(1/x),x) and integrate(cos(1/x),x) were

integrate(sin(1/x),x) = (    x - REC3(x,1,ln(x))    ) * sin(1/x) +
                        ( ln(x) - RECsc(x,1,ln(x))  ) * cos(1/x) .

integrate(cos(1/x),x) = ( -ln(x) - RECcs(x,1,ln(x)) ) * sin(1/x) +
                        (    x - REC3(x,1,ln(x))    ) * cos(1/x) .

  where REC3, RECsc and RECcs are 3 itself-recursive functions of integration
   that never terminate.

REC3(x,M,L)  = ( - 1/M ) *
             { ((L+1/M)/(x^M) + (1/(M+1))*REC3(x,M+2,L+(1/M)+1/(M+1)) }.

RECsc(x,M,L) = ( 1/(M*(M+1)) ) *
             {   ((L+(1/M)+1/(M+1))/(x^(M+1)) - RECsc(x,M+2,L+(1/M)+1/(M+1)) }.

RECcs(x,M,L) = ( 1/(M*(M+1)) ) *
             { - ((L+(1/M)+1/(M+1))/(x^(M+1)) - RECcs(x,M+2,L+(1/M)+1/(M+1)) }.

You can expand manually step-step these simplified integrals and you will see
 them that they never terminate the computation.

By the moment, i don't know if these 3 recursive functions REC?? have exact
 numeric solutions for sin(1/x) & cos(1/x) or not, have functional, polynomial
 or serial representations or not, to become them into primitives functions.

Until now, it's very harder to solve and not recommend to follow them.

I'm near to solve it using the simplification by the substitution method
 trying to integrate cos(x)/(x^2) dx later due to

   integrate(cos(1/x),x,0,2/%pi) = - integrate(cos(t)/(t^2),t,%pi/2, inf)

   Sincerely, J.C.Pizarro