Re: [Axiom-developer] Bug in complexForm : complexForm(log(%i) - log(-%i)) -> 0

[Martin Rubey]

Dear Vladimir,

Vladimir Bondarenko writes:
 > AXIOM ->         complexForm(log(%i) - log(-%i))
 > 
 >                  0

could you please file this as a bug on issuetracker? The problem is in the
operation "argument":

(8) -> argument(%i)

        %pi
   (8)  ---
         2
                                                     Type: Expression Integer
(9) -> argument(-%i)

        %pi
   (9)  ---
         2
                                                     Type: Expression Integer

the latter should be -pi/2, of course. The problem is in gaussian.spad,
COMPCAT:

       if R has TranscendentalFunctionCategory then
         half := recip(2::R)::R

         if R has RealNumberSystem then
           atan2loc(y: R, x: R): R ==
               pi1 := pi()$R
               pi2 := pi1 * half
               x = 0 => if y >= 0 then pi2 else -pi2

               -- Atan in (-pi/2,pi/2]
               theta := atan(y * recip(x)::R)
               while theta <= -pi2 repeat theta := theta + pi1
               while theta >   pi2 repeat theta := theta - pi1

               x >= 0 => theta      -- I or IV

               if y >= 0 then
                   theta + pi1      -- II
               else
                   theta - pi1      -- III

           argument x == atan2loc(imag x, real x)

         else
           -- Not ordered so dictate two quadrants
           argument x ==
             zero? real x => pi()$R * half
             atan(imag(x) * recip(real x)::R)

so the problem occurs if R does not have "RealNumberSystem":

(15) -> argument(-%i*1.0)

   (15)  - 1.5707963267 948966192

since Float does have RNS, but INT does not. I'm not quite sure what the
condition should be. OrderedSet?

Martin