powerseries

On Wed, July 11, 2007 at  8:44AM, Barton Willis wrote:
> I think the function %piargs (in trigi.lisp) is buggy

this seems to fix it:

*** trigi.lisp	19 Apr 2007 12:47:19 -0400	1.27
--- trigi.lisp	11 Jul 2007 11:30:13 -0400	
***************
*** 631,637 ****
    (cond ((and (integerp (car x)) (integerp (cdr x))) 0)
  	((not (mevenp (car x))) 
  	 (cond ((null ratcoeff) nil)
! 	       ((alike1 (cdr x) '((rat) 1 2))
  		(power -1 (add ratcoeff -1//2)))))
  	((or (alike1 '((rat) 1 6) (setq x (mmod (cdr x) 2))) (alike1 '((rat) 5 6) x)) 1//2)
  	((or (alike1 '((rat) 1 4) x) (alike1 '((rat) 3 4) x)) (div (power 2 1//2) 2))
--- 631,637 ----
    (cond ((and (integerp (car x)) (integerp (cdr x))) 0)
  	((not (mevenp (car x))) 
  	 (cond ((null ratcoeff) nil)
! 	       ((and (integerp (car x)) (alike1 (cdr x) '((rat) 1 2)))
  		(power -1 (add ratcoeff -1//2)))))
  	((or (alike1 '((rat) 1 6) (setq x (mmod (cdr x) 2))) (alike1 '((rat) 5 6) x)) 1//2)
  	((or (alike1 '((rat) 1 4) x) (alike1 '((rat) 3 4) x)) (div (power 2 1//2) 2))


(%i2) powerseries(1/(1+x+x^2),x,0)$

(%i3) rectform(%);
Is  x  positive or negative?
pos;
(%o3) 'sum((-sin(%pi*(-i1-1)/3)*(-1)^i1/sqrt(3)-sin(2*%pi*(-i1-1)/3)/sqrt(3))
            *x^i1,i1,0,inf)
       +%i*'sum((cos(%pi*(-i1-1)/3)*(-1)^i1/sqrt(3)
                 +cos(2*%pi*(-i1-1)/3)/sqrt(3))
                 *x^i1,i1,0,inf)
(%i4) subst(inf=5,%);
(%o4) -x^4+x^3-x+1

(%i5) declare(y, integer);
(%o5) done

(%i6) cos(2*y*%pi/3);
(%o6) cos(2*%pi*y/3)

(%i7) cos(2*y*%pi);
(%o7) 1

(%i8) cos(y*%pi);
(%o8) (-1)^y

(%i9) cos(3*y*%pi);
(%o9) (-1)^y