From: hebisch@antispam.uni.wroc.pl   
      
   Richard Fateman  wrote:   
   > I suspect that part of the problem in getting   
   > back via differentiation may be that antiderivatives   
   > computed using any of the purely algebraic methods   
   > are not cognizant of branch cuts or discontinuities   
   > at all. I think this analysis-related stuff is deserving of   
   > serious consideration, but demanding that some Risch-based   
   > procedure produce an optimal (e.g. continuous if possible)   
   > antiderivative may not be a "debug this"   but "can we make   
   > another pass over the answer".   
   >   
   > It's not that the Risch algorithm(?) is too hard, but that   
   > it is not computing what you want...   
   >   
   > Returning to the integrand via differentiation putting you   
   > on the wrong side of a branch cut?  Whose fault is that if   
   > differential algebra doesn't even have that concept?   
      
   Actually, if kernels in the input are independent, then   
   algebraic approach has no problem with differentiating   
   the result.  Namely, independence of kernels implies that   
   for each choice of branches we get izomorphic differential   
   field.  Risch algorithm produces answer from ingredients   
   in the field and this anwer as expression should differentiate   
   back to original expression.  At algebraic level various   
   substititions are valid because they are izomorphizms   
   of differential fields.  So, why troubles?  First, one   
   have to be careful to undo substitutions.  Next, practical   
   implementations take shortcuts compared to Risch algorithm   
   and those shortcuts may introduce extra terms, outside   
   differential field in question.  Also, there is problem   
   with representing differential fields.  In FriCAS   
   there is no explicit representation of differential fields.   
   Instead, kernels appearing in a function implititely   
   determine the field.  But this interactcs badly with   
   automathic simplifications.  For example, we have   
   sqrt(3/4) and this gets simplified to sqrt(3)/2.   
   Lovers of principal branches consider this valid,   
   but such change destroys structure of fields.   
   So really trouble were due to imperfect implementation.   
   Let me add that _most_ brach problem in FriCAS were   
   due to non-Risch parts of the integrator.   
      
   Of couse dependent kernels cause trouble for algebraic   
   approach, but this is more on pragmatic level.  Namely, should   
   integrator try to solve potentially hard (unsolvable)   
   problems which actually have little to do with integration.   
   ATM in case of dependent kernels FriCAS assumes that   
   branches will simplify.  But in principle other approaches   
   are possible.   
      
   Anyway, the example that started this disscusin, namely   
      
   acos(x^2 - sqrt(1 - x^2))   
      
   has independent kernels.   
      
   --   
                                 Waldek Hebisch   
      
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