Am 06.12.2014 um 21:01 schrieb Aaron Meurer:
Something that I'm not sure about with representing functions as
multivalued is, how do you represent arbitrary Riemann surfaces.
Another question is computational. How do you compute the surfaces in
general (say even for a limited class of expressions, like algebraic
functions), and how do you make cuts in a consistent manner?
I do not thing that consistent cuts are a well-defined concept in the
first place.
Consider having sin 2x and sin x in the same expression. (Or something
equivalent in log and exp.)
Things become more, er, interesting with more free variables (x, y, z, ...)
To catch all equalities:
- Parameterize them all with an integer.
- For n integer parameters, search Z^n (exploiting periods).
This might turn out to be too slow to be useful.
OTOH it might help with some simplifications.
The only thing I can think of is to write everything in terms of exp()
and log() and parameterized integers,
Can everything be broken down into this?
I'd have expected that not, since the cardinality of the set of
functions isn't countable, but maybe it's just that all practically
relevant expressions can be broken down that way, or maybe there's some
result that indeed everything multivalued can be written that way.
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