[GiNaC-list] Re: GiNaC-list Digest, Vol 29, Issue 1

[Richard B. Kreckel]

Hi!

Richard B. Kreckel wrote:
>>                                   Of course I can't oversee the
>> consequences of a change, it would be convenient to code decisions based
>> on the argument in a uniform way though.
> 
> Let's see. atan2 appears in LIA-2 and LIA-3, though not in LIA-1. It 
> isn't defined directly, but one finds for instance the formula in LIA-3: 
> ln(z)=ln(|z|)+I*atan2(y,x) for z=x+I*y. This implies to bring the branch 
> cut in alignment with the branch cut of ln(x+I*y). The latter is defined 
> as continuous with the second quadrant of the x-y plane (c.f. C99, 
> 7.3.3.2: "[...] implementations shall map a cut so the function is 
> continuous as the cut is approached coming around the finite endpoint of 
> the cut in a counter clockwise direction. (Brnach cuts for the functions 
> specified here have just one finite endpoint.)" As I said, this branch 
> cut specification for ln(z) is consistent with the other standards.
> 
> IINM this implies that atan2(0,-4) should be +Pi, not -Pi.

The old definition of atan2(0,-4)==-Pi did not only disagree with C99, 
it also caused real inconsistencies:

     ex z = -4;
     cout << log(z) << endl;  // log(4)+I*Pi
     cout << log(z).imag_part() << endl;  // -Pi

I've fixed that on both active branches in CVS.

> Unless somebody is going to hold me back I'm going to fix this in GiNaC 
> RSN. I'll also consider some minor other issues like atan2(I*1.1,1.1) 
> throws an exception but atan2(I,1)->atan2(I,1) (both arguments of atan2 
> must be real, full stop.)

Restricting atan2(y,x) to real arguments would be more restrictive than 
is needed. A suitable analytic continuation of atan2(y,x) to complex 
arguments y and x can be found in some other systems, too: 
-I*log((x+I*y)/sqrt(x^2+y^2)). As a result, more values are accepted, now.

Cheers
    -richy.
-- 
Richard B. Kreckel
<http://www.ginac.de/~kreckel/>