complex infinity

[uunet!cwi.nl!Dik.Winter]

Some additional comments.
 > Infinity is even more important in complex functions than in real functions.
True.
 > 
 >        I think the appropriate default way of thinking about them from the
 > point of view of computer representation is that any complex number with 
 > two infinite or one infinite and one finite component is a complex infinity
 > and represents the point on the top of the complex ball.
 > The signs of the infinities and numerical value of the finite component,
 > if any, may tell you something useful about how you got to the infinity
 > or they may just be noise, just as in the case of the sign bit of
 > real infinity.
I have to disagree a bit.  The issue of just one or more than one complex
infinity is similar to the issue of just one or more than one real infinity.
In most cases having one infinity is adequate, but there are a number of
cases where you really want more than one infinity.  We need however some
consistency with the real case.  I think in the complex case, if you want
to distinguish infinities, it is more important to know the direction of
the infinity.  So you really want the infinities to be presented in polar
coordinates!  And so that is not a separate issue.  Assuming carthesian
coordinates for infinities, what is the distinguishing feature between
(1.0, inf) and (2.0, inf)?  Really nothing!  Like in the real case you would
want that the multiplication of an infinity by a positive real would not
change the infinity, but if you maintain carthesian coordinates for infinities
this will not hold.  So I think a more proper representation of infinities
would be in polar coordinates.  But this opens a complete can of worms of
course!  What about the different representations of (0.0, 0.0)!
So do you want to distinguish between (1.0, +0.0) and (1.0, -0.0), and if
so why?

In the real case a point can be made in a number of occasions for the IEEE-754
position on signed zeros and signed infinities (the atan2 functions being
one of those, atan2(0.0, -1.0) depends on the sign of 0.0).  In the complex
case it is not so simple.  I think that if your representation is carthesian
that you should allow only one representation for zero and infinity.  Only
with a polar representation can you adequately represent the different values.
But what if you were going to define a function atan2 for complex operands?
Is it possible to get a consistent behaviour between real aran2 and complex
atan2?

dik
--
dik t. winter, cwi, kruislaan 413, 1098 sj  amsterdam, nederland
dikacwi.nl