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Desirable Refinements

  1. Some people don't like working over the complex plane. We can avoid factoring over C and factor only to a real form; this gives factors like , which significantly complicate the actual integration part of the algorithm (instead of simply using the rule (note this gets the correct logarithmic form in the limit as )), we have to use integration by parts and arctans. It's all do-able, of course, just more complicated.
  2. It's not obvious here if we can avoid factoring over the real numbers. But some problems (e.g. integrating the derivative of ) will not require a final answer to be expressed using any algebraic extensions of the rational numbers. It would be nice if there was an algorithm which avoided these extensions when it could get away with it.
  3. What do we do when can't be factored? If the roots are all simple then there isn't much problem: we write the answer as, e.g.

    and for many purposes this form answers admirably, if we can compute (which we can---it's just a residue, and in the general case it's just ---for simple roots).

Rob Corless
Thu Nov 16 13:46:20 PST 1995