Waldek Hebisch schrieb:   
   >   
   > clicliclic@freenet.de wrote:   
   > >   
   > > I don't recall hearing any advocates here on sci.math.symbolic, and   
   > > am unaware of problems caused by the use of numerical   
   > > differentiation to check symbolic derivatives - when such a check   
   > > fails, the symbolic result must be wrong provided the numerics is up   
   > > to the task. In practice, I am using the fast numerical integration   
   > > available for plotting on Derive to check dubious portions of   
   > > antiderivatives.   
   >   
   > You wrote "provided the numerics is up to the task".  In general   
   > you need to correctly handle branches is numerics.  Simply   
   > "plugging in" expressions into numeric evaluator will not   
   > produce correct braches for the problem: brach cuts in numeric   
   > definitions tend to produce spurious discontinuities.   
      
   ... and accuracy control allows one to make any numerical evaluation   
   as stable as desired.   
      
   > >   
   > > I don't get the point of this. Complex functions - be they atomic or   
   > > composite expressions - must often be rendered single-valued to be   
   > > meaningful, and everybody concerned (Ahlfors and a reader; a user   
   > > and FriCAS) must agree about the branch assigned to them. When   
   > > selecting a 'principal' branch for a complex function, the   
   > > corresponding real function should preferably be subsumed and   
   > > unnecessary discontinuities be avoided - otherwise the choice is   
   > > mostly by convention, including principles like 'CCC'   
   > > (Counter-Clockwise Continuity). So there are preferences and there   
   > > is freedom. Whenever one's mathematics depends on the choice, it has   
   > > to be made clear what branch one is using.   
   >   
   > You ignore the fact that symbolic computations usualy do not   
   > depend on choice of branches: from symbolic point of view to   
   > branches are indistingushable.  Once correct symbolic result   
   > is obtained, to solve analytic problem one may need to   
   > determine branches.  Choice of branches is specific to given   
   > problem and frequently requires information not present in   
   > symbolic version.  Of course, once you have concrete analytic   
   > problem you need clear choice of branches.  But usually such   
   > choice can be separated from other considerations.  In particular   
   > you can produce useful symbolic results without caring about   
   > branches.   
      
   Would z*dif(asec(z), z) = 1/sqrt(z^2-1)) or log(z) + log(w) = log(z*w)   
   be such a "useful symbolic result"?   
      
   While many symbolic manipulations involving complex functions do not   
   depend on the choice of branches, they may mix those branches so that   
   the results can be expected to apply only if the functions are viewed as   
   multi-valued - as involving all their branches simultaneously, of which   
   there are often infinitely many. This view is obviously incompatible   
   with a context-independent unambiguous evaluation of the functions, and   
   with the possible exception (?) of Axiom/FriCAS, I know of no computer   
   algebra system that takes it. Rather, systems prefer to make complex   
   functions single-valued symbolically as well as numerically, restricting   
   the atomic functions to immutable standard branches and defining the   
   branches of arbitrary expressions through functional composition.   
      
   This restriction entails both a loss and a gain. On the downside, many   
   functional relations of multi-valued complex analysis cease to hold   
   without qualification, and composite functions may easily (depending on   
   user attention on input, system attention on output) be specified such   
   that they carry convoluted branch structures; on the upside, the   
   restriction allows symbolic results to be checked numerically by the   
   user - in particular visually after plotting - in the manner familiar   
   from real analysis, and systems may also use it internally to check and   
   supplant their symbolic computations - for instance, they may accept   
   numerical in place of undecidable symbolic zero-equivalence (cf.   
   Mathematica).   
      
   You seem to believe the gain not to be worth the loss, but I don't see   
   how the present FriCAS can be taken to support this belief, nor do I see   
   interesting unique capabilities that could result from pursuing it.   
      
   > >   
   > > Each of the complex functions of a CAS, on the other hand, comes   
   > > with a specific branch built into the numerics, and its symbolics is   
   > > hopefully designed to comply - no provision is made for the user to   
   > > specify a new branch choice for any atomic or composite function, or   
   > > for the numerics and symbolics to accept and act on such a   
   > > specification. This leaves the preinstalled branches and their   
   > > functional composition as the only unambiguous branch assignment   
   > > practical for arbitrary expressions.   
   >   
   > Once you are dealing with arbitrary expressions numeric branches   
   > frequently are bogus.  You simply can not use "arbitrary expression"   
   > as useful definition of branch behaviour.  Simply, branch information   
   > must be carried "out of band".  If you want symbolic representation   
   > one possibility is to specify domain and choice at a single point   
   > (frequently just specifying value is enough to identify branch).   
   > If expression is analytic and domain is simply connected, then   
   > this uniquely determines the function.  Clearly, significant   
   > effert will be needed to make approach like this practical.   
      
   Pie in the sky.   
      
   Branches "can" be assigned through functional composition because one is   
   free to do so. Though 'nicer' choices will often exist, this is "useful"   
   because an unambiguous evaluation is imparted to any expression without   
   reference to separate information. Your "must" does therefore only   
   indicate a personal viewpoint, not a mathematical necessity. And your   
   separate step of analytic continuation from a specified germ could be   
   performed equally well when symbolic results systematically reflect   
   input branch cuts; the ambiguity characteristic of alternative branch   
   assignments would then enter through this optional step alone, while the   
   gain outlined would not be sacrificed without need.   
      
   All this will be above the head of the great majority of users with   
   nothing but real functions of a real variable on their mind. Yet such a   
   user may well happen to cobble together a differential equation which   
   leads to a simple integral whose solution by FriCAS 1.2.6 doesn't hold   
   everywhere (because the standard branches cause the complex plane to   
   break into incompatible pieces of which FriCAS adopts one for the entire   
   plane), and whether his solution is valid (cf. Timofeev Chapter 1,   
   Example 64 for a>0, x>0) or invalid (ditto for a<0, x<0) is a matter of   
   luck. The nuisance caused by having to find out that the solution   
   returned doesn't satisfy the differential equation symbolically, and   
   then perhaps trying to determine numerically if and where in a   
      
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