On 4/27/2013 10:58 AM, firstname.lastname@example.org wrote: > > Richard Fateman schrieb: >> >> It is perhaps worth noting that algorithms for antidifferentiation >> which are based on differential algebra (e.g. Risch), and perhaps >> much else (like typical calc books), do not deal with ANALYTIC >> properties, except by coincidence. > > Even the people studying calculus books (and more so the users of Risch > integrators) would find life easier if their antiderivatives held on the > entire real line (or on the entire complex plane, as the case may be) > and could be verified by differentiation (if not in the given CAS, then > at least, in principle, on paper). No other "analytic" property is > required of them.
That's pretty much analytic, in my book. Reference to the real line.
While you may prefer expressions that are continuous etc, that does not fall out automatically from Risch algorithm etc. The problem has been studied though, eg. by Rioboo.
The other problem, of testing for two expressions being identical is obviously reducible to testing if their difference is zero, which is known to be (in the general case) undecidable. A useful heuristic is to pick random numbers(s) for x, etc. If the result is not zero, maybe the expression is not identically zero.
For certain classes of expressions (rational functions in particular) the problem is solvable, but introducing log, I, Pi, sqrt, may be enough to break it. (results of Daniel Richardson).
CAS users (and sometimes texts!) casually introduce expressions involving square root into problems without specifying WHICH square root, or even recognizing that there is an issue.
But then then when they get some answer they insist that the answer is written in terms of the WRONG square root. (Well, that is an oversimplification, but you should get the idea..)