Richard Fateman schrieb:   
   >   
   > On 7/24/2016 3:48 PM, Nasser M. Abbasi wrote:   
   > > I tried this on Maple, Mathematica, MuPad (part of Matlab) and   
   > > Matlab.   
   > >   
   > > Asked for maximum of I and 1.0, where I=sqrt(-1).   
   > >   
   >   
   > [...]   
   >   
   > So what to do.   A symbolic system could do one of a number of   
   > things.   
   >   1. Something more or less bogus like Matlab.   
   >   2. Give an error message   
   > and stop computing (perhaps wait for a user to supply an answer?   
   > perhaps berate the user in some way by appealing to the Algebra   
   > course he/she never took?)   
   >   3. Leave the expression unevaluated as max(W,V).   
   > (hoping that it might perhaps be multiplied by zero and not matter???)   
   >   4. Exercise some judgment of the form  "the imaginary part is down   
   > in the roundoff, let's keep computing." One way to handle this   
   > is to drop it, make a note of the occurrence and alert the user at the   
   > end of the computation that a certain number of such transformations   
   > were done.  E.g. plotting a mathematically real function f(x) which   
   > numerically turns out to have some tiny imaginary parts.   
   >   5. Make it possible to do any of the choices above, or maybe   
   > some others?   
   >   
   > Maxima does 3  by default, but could be instructed to do any of   
   > the other choices by modifying the simplification of max().   
   >   
   > Mathematica gives a warning message, but then does 3.  You   
   > can redefine Max in Mathematica if you first Unprotect it.   
   >   
      
   Derive 6.10 also just leaves MAX(#i, 1) untouched under exact as well as   
   approximate (e.g. ten decimal digit) evaluation. No error message is   
   generated. And because all Derive numbers are exact rationals, MAX(0.0 +   
   1.0*#i, 1.0 + 0.0*#i) means exactly the same as MAX(#i, 1).   
      
   Jeff's example suggests that the limit LIM(MAX(-2 + eps*#i, 1), eps, 0)   
   differs from MAX(-2, 1) in MATLAB. So if Matlab's extension of MAX works   
   properly for real arguments, it will be discontinuous for complex ones.   
   It would have to return MAX(-ABS(-2 + eps*#i), 1) to be continuous here.   
   Maybe Matlab's function arguments are typed and we are dealing with   
   separate functions here: MAX(real, real) and MAX(complex, complex). In   
   such a world, -2 + eps*#i tending to -2 would remain complex and could   
   never become real ...   
      
   In mathematics, and therefore in Computer Algebra, however, the real   
   numbers are a subset of the complex numbers and MAX(ABS(x), ABS(y)) =   
   MAX(x, y) is mathematically wrong for certain real arguments. If someone   
   is here interested in MAX(ABS(x), ABS(y)) for real or complex arguments,   
   he would be required to write just this. For repeated use he could   
   define his own function MAXABS(x,y) := MAX(ABS(x), ABS(y)) and then   
   write simply MAXABS(x,y).   
      
   A mathematically sensible extension to the complex numbers could be   
   component-wise application: MAX(u, v) := MAX(RE(u), RE(v)) +   
   #i*MAX(IM(u), IM(v)). I suppose some systems provide a component-wise   
   FLOOR and ROUND already. A component-wise ABS(z) := ABS(RE(z)) +   
   #i*ABS(IM(z)) that is compatible with such a component-wise MAX would   
   collide with the standard mathematical notion of the absolute value of   
   complex numbers, however, so a different name would be needed ...   
      
   Martin.   
      
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