[Axiom-developer] [#47 complexForm(log(%i) - log(-%i))]

[wyscc]

Changes 
http://page.axiom-developer.org/zope/mathaction/47ComplexFormLogILogI/diff
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Martin: 

Thanks for pointing out my error. In coding 'myArgument', I was not aiming for 
a general definition. I was experimenting to see if there is need for 
'RealNumberSystem' or 'OrderedRing'. 
As you proposed, it seems the error in 'myArgument' may be fixed by a two 
argument version of 'atan' (which only exists for 'Float' and 'DoubleFloat' 
currently). The usual sign convention will take care of the case of inputs 
involving variables. But I am not convinced one way or the other that 
'OrderedRing' is needed.

\begin{axiom}
myhalf: ()-> EXPR INT
myhalf()==recip(2::EXPR INT)::EXPR INT
pi1:= pi()$(EXPR INT)
pi2 := pi1 * myhalf()
myatan2(x:EXPR INT, y: EXPR INT): EXPR INT ==
  x = 0 => if y >= 0 then %pi else -%pi
  -- Atan in (-pi/2,pi/2]
  theta := atan(y * recip(x)::EXPR INT)
  -- may create infinite loop if variables are used in input
  -- in any case, why would atan give a result outside (-pi/2, pi/2]?
  -- while theta <= -pi2 repeat theta := theta + pi1
  -- while theta > pi2 repeat theta := theta - pi1
  x >= 0 => theta      -- I or IV
  if y >= 0 then
      theta + pi1      -- II
  else
      theta - pi1      -- III
myatan2(1,-1)
myatan2(y,x)
myArgument: Complex EXPR INT -> EXPR INT
myArgument(x) == 
  zero? real x =>
    imag(x) > 0 => pi2
    imag(x) < 0 => - pi2
    error "myArgument not defined at (0,0)"
  myatan2(real x, imag x)
myArgument(-%i)
myArgument(%i)
myArgument(3+5*%i)
myArgument(3-5*%i)
myArgument(-1+%i)
myArgument(2 - x*%i)
myArgument(0::Complex EXPR INT)
\end{axiom}

William
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