Infinities and Singularities

Sergey B Kirpichev edited this page Nov 22, 2013 · 2 revisions
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Preamble

While beginning to work with the codebase I have stumbled a couple of times over the aforementioned problem. A situation occured in bug 1321; also bug 360 is directly related. The "official" ticket is 2200. You can find there also a link to previous discussion, though under a non-ideal heading.

Epistemology

There are three kinds of objects involved: expressions, "limit directions" and "limit result descriptions".

Expressions are the things we want to take limits of, like cos(x), 1/ x^2, exp(z), ...

Limit directions describe how the limit is to be taken. In R (reals), this generally is a point and a side of approach, or +oo or -oo. In general topological spaces limit directions can be described by filters.

Other common filters are: a point of C (complex plane), approached from all sides. This is the definition of limit of normed spaces and hence generalises. Also ComplexInfinity is a well-defined limit direction.

Limit result descriptions are the answers of limits. These can be very complicated, if a lot of information is to be conveyed. For example:

  • limits can exist as ordinary numbers, this is the base case (limit(sin(x), x, 0) == 0 for example)
  • limits taken in R can be +oo or -oo in a well-defined sense, e.g. limit(x**2, x, oo)
  • limits such as limit(sin(x), x, oo) have to be described in a more complicated way. The result could say "point of accumulation for any x in [-1, 1]".
  • limits of meromorphic functions in C can be classified as a number, a pole of order n (for some n > 0), or essential singularity
  • in general, not much can be said. Even a rational function of two variables can have limits that cannot usefully be interpreted as a number or infinity.

Discussion

Much confusion arises, I think, because objects of the three different classes are often described by the same names:

  • a number can be seen as an expression and as a limit description, and also as a limit direction at least two different ways (over C and over R)
  • oo can be seen as both a singularity description and as a limit direction
  • ditto for zoo

Moreover +-oo can be seen as two points of an extended real line (topologically its end-compactification). Similarly zoo can be seen as a point on the riemann sphere.

Finally, we typically want to be able to do at least some arithmetic with infinities, to write things such as oo+1 == oo. SympyCore calls this extended number system.

Operations with filters

We have (in principle) the following operations, for f(z), g(z, y) any functions of complex variables, F, G any filters

  • limit(f, F).
  • f(F) =: filter generated by {f(U) | U \in F}
  • g(F, G) =: filter generated by {f(U, V) | (U, V) \in FxG}

The latter two allow many "arithmetic" operations to be performed on filters, and the result are often of the expected form.

Limit directions as filters

So we have the following commonly used filters (I find it easier to think in terms of filter bases):

for x in R:

RB(x) = {filter generated by [x, x+1/n) for n in N}
LB(x) = {filter generated by (x-1/n, x] for n in N}
B(x)  = LB(x) + RB(x) = {filter generated by (x-1/n, x+1/n)}

For z in C: CB(z) = {filter generated by {y in C: |z-y| < 1/n}, n in N} P(z) = {filter generated by {z}} = {all subsets of C containing z}.

oo = {filter generated by (x, oo) for x > 0}
-oo = -1 * oo = {filter generated by (-oo, x) for x < 0}

zoo = {filter generated by {y in C: |y| > n}, n in N}

Why filters are not limit result descriptions

limit(f, F) does not return a filter: it usually returns a number. Indeed limit(f, F) = c iff for all U in CB(c), there exists V in f(F) with U \subset V.

Hence it would be desirable to find an equivalence relation on the set of all filters, such that

  • Distinct principal ultrafilters are not equivalent.
  • All filters converging to the same point are equivalent.
  • A non-convergent filter is not equivaelent to a convergent filter.
  • The equivalence class of f(F) can be computed in many interesting cases.
  • Arithmetic on filters (as defined above) descends in a useful way to the equivalence classes.

Please do tell if you know such an equivalence relation :-).