Albert Rich  wrote:   
   >   
   > log(z^2) is not equal to 2*log(z) over half the complex plane.   
      
   That depends on definitions.  Apparently you believe that log(z^2)   
   and 2*log(z) are differentiable.  Classic theorem of complex analysis   
   says that two functions defined and having complex derivative on a   
   connected open set which are equal in a neighbourhood of a point   
   agree on whole set.   
      
   >   
   > It seems to me if indeed the Risch Algorithm cannot be trusted to produce   
   analytically valid antiderivatives over the complex plane, a computer algebra   
   system's Integrate command should first try using elementary methods that do.   
   Then iff that fails,    
   resort to the Risch Algorithm AND warn users the result may not be valid over   
   the complex plane.   
   >   
   > As Rubi demonstrates, rule-based elementary methods can produce optimal,   
   analytically valid antiderivatives for a large class of integrands.   
      
   You seem to believe in existence of discontinous differentiable   
   functions which hard to reconcile with "analytically valid".   
   Both theoretically and practically it seems better to say   
   that we deal with continuous branches of analytic functions   
   (possibly defined on some Riemann surface).  In case of   
   logarithms possible branches differ by constans and we can   
   choose them in way so that 2*log(x) = log(x^2) and more general   
   log(f(x)*g(x)) = log(f(x)) + log(g(x)).  Once we have set   
   of branches satifying those relations symbolic result does   
   not depend on specific choice of branches, that is other   
   choice of branches satisfying relations leads to the   
   same formulas.  FriCAS assumes such choice of branches   
   because it is what many users want, and users needing other   
   choice can add explicit constants to get effect corresponding   
   to different branches.   
      
   --   
                                 Waldek Hebisch   
      
   --- SoupGate-Win32 v1.05   
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