Checking equality with ComplexInfinity and testing whether a function is (mathematically) undefined

Question

This seems ridiculously basic, but I cannot find anywhere explaining how to do it.

If you evaluate certain undefined functions like 1/0 or Tan[\[Pi]/2], Mathematica evaluates to ComplexInfinity. For some reason, however, the following expression does not act as expected:

1/0 == ComplexInfinity

This results in

ComplexInfinity == ComplexInfinity

rather than the expected

True

So the question is, how do you test whether a function evaluates to ComplexInfinity, or better yet, how do you test if a function is (mathematically, not programmatically) undefined?

Answer 1

To understand why ComplexInfinity == ComplexInfinity, let's look at how Mathematica handles infinities. All of Infinity, -Infinity, ComplexInfinity are just displayed forms of DirectedInfinity:

In[1]:= FullForm /@ {Infinity, -Infinity, ComplexInfinity}
Out[1]= {DirectedInfinity[1], DirectedInfinity[-1], DirectedInfinity[]}

DirectedInfinity[z] represents an infinite quantity in the direction of z on the complex plane. ComplexInfinity is DirectedInfinity[]; i.e., an infinite quantity with an unknown direction. The reason it cannot be compared is that the direction is unknown.

---

If you need to test for ComplexInfinity in particular, you can use

something === ComplexInfinity

=== is a structural test that, unlike ==, does not take any mathematical meaning into account.

---

Finally, for your purpose, I think NumericalQ (your answer) is a very good solution.

Answer 2

I found an easy way to do this, simply test the value of the function with NumericQ to determine if it evaluates to a number or not. I would still be interested to know what sort of object or entity ComplexInfinity is that I can't test whether it is equal to itself with ==.