[Rd] Add to Documentation of atan2.

[Ben Bolker]

   Can you dig through into the code, see what's going on, and suggest a 
documentation patch?  To get you started, the code for the complex 
version of atan2 is in

https://svn.r-project.org/R/trunk/src/main/complex.c

z_atan2 is at line 669 (the first argument is a pointer to the result, 
args 2 [csn] and 3 [ccs] are pointers to the arguments of atan2())

  In generic cases the computation is

	dr = catan(dcsn / dccs);
	if(creal(dccs) < 0) dr += M_PI;
	if(creal(dr) > M_PI) dr -= 2 * M_PI;

where dcsn, dccs are converted versions of the args.

catan() is *either* taken from system libraries or is defined at line 489.

   On my system (Ubuntu), 'man 3 catan' gives documentation on the 
function, and says "The real part of y is chosen in the interval 
[-pi/2,pi/2]" - but that _could_ be system-dependent.

    cheers
    Ben Bolker

On 5/18/21 10:39 AM, Jorgen Harmse via R-devel wrote:
> The current documentation says that atan2(y,x) is the angle between the x-axis and the vector from the origin to (x,y), but what does this mean when x & y are complex? The function seems to pick theta with Re(theta) between -pi and pi and with tan(theta) (approximately) equal to y/x, but that leaves 2 (sometimes 3) options, and there must be a set (branch region with 3 real dimensions?) on which the function is discontinuous. Please add details.
> 
> Even for real inputs, it might help to spell out the behaviour on the negative x-axis. It mostly matches the branch-cut rules for the other functions, but atan2(0,0)==0 is a unexpected.
> 
> I also suggest ‘See Also’ links from trigonometric functions to hyperbolic functions and from hyperbolic functions to exponential & logarithmic functions.
> 
> Regards,
> Jorgen Harmse.
> 
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>> R.version.string
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> [1] "R version 4.0.4 (2021-02-15)"
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