[Numpy-discussion] arctan2 with complex args

[Timothy Hochberg]

On 4/30/07, David Goldsmith <David.L.Goldsmith at noaa.gov> wrote:
>
> lorenzo bolla wrote:
> > 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!
> >
> This is the foreseeable use case described by Anne.
>
> In any event, I stand not only corrected, but embarrassed (for numpy):
>
>     Python 2.5 (r25:51918, Sep 19 2006, 08:49:13)
>     [GCC 4.0.1 (Apple Computer, Inc. build 5341)] on darwin
>     Type "help", "copyright", "credits" or "license" for more information.
>      >>> import numpy as N
>      >>> complex(0,1)
>     1j
>      >>> N.arctan(1)
>     0.78539816339744828
>      >>> N.arctan(complex(0,1))
>     Warning: invalid value encountered in arctan
>     (nannanj)
>
> I agree that arctan should be implemented for _at least_ *one* complex
> argument...


It is, you should look at that error a bit more carefully...

-tim

(hint what is arctan(0+1j)?)


DG
> >
> >    1. 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
> > <mailto:peridot.faceted at gmail.com>> wrote:
> >
> >     On 29/04/07, David Goldsmith <David.L.Goldsmith at noaa.gov
> >     <mailto: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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> >     http://projects.scipy.org/mailman/listinfo/numpy-discussion
> >
> >
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-- 

//=][=\\

tim.hochberg at ieee.org
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