[Numpy-discussion] arctan2 with complex args

[lorenzo bolla]

me!
I have two cases.

   1. I need that arctan2(1+0.00000001j,1-0.000001j) gives something
   close to arctan2(1,1): any decent analytic prolungation will do!
   2. if someone of you is familiar with electromagnetic problems, in
   particular with Snell's law, will recognize that in case of total
   internal reflection<http://en.wikipedia.org/wiki/Total_internal_reflection>the
wavevector tangential to the interface is real, while the normal one
is
   purely imaginary: hence the angle of diffraction is still given by
   arctan2(k_tangent, k_normal), that, as in Matlab or Octave, should give pi/2
   (that physically means no propagation -- total internal reflection, as
   said).

L.

On 4/30/07, Anne Archibald <peridot.faceted at gmail.com> wrote:
>
> On 29/04/07, David Goldsmith <David.L.Goldsmith at noaa.gov> wrote:
> > Far be it from me to challenge the mighty Wolfram, but I'm not sure that
> > using the *formula* for calculating the arctan of a *single* complex
> > argument from its real and imaginary parts makes any sense if x and/or y
> > are themselves complex (in particular, does Lim(formula), as the
> > imaginary part of complex x and/or y approaches zero, approach
> > arctan2(realpart(x), realpart(y)?) - without going to the trouble to
> > determine it one way or another, I'd be surprised if "their"
> > continuation of the arctan2 function from RxR to CxC is (a. e.)
> > continuous (not that I know for sure that "mine" is...).
>
> Well, yes, in fact, theirs is continuous, and in fact analytic, except
> along the branch cuts (which they describe). Formulas almost always
> yield continuous functions apart from easy-to-recognize cases. (This
> can be made into a specific theorem if you're determined.)
>
> Their formula is a pretty reasonable choice, given that it's not at
> all clear what arctan2 should mean for complex arguments. But for
> numpy it's tempting to simply throw an exception (which would catch
> quite a few programmer errors that would otherwise manifest as
> nonsense numbers). Still, I suppose defining it on the complex numbers
> in a way that is continuous close to the real plane allows people to
> put in almost-real complex numbers and get out pretty much the answer
> they expect. Does anyone have an application for which they need
> arctan2 of, say, (1+i,1-i)?
>
> Anne
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