Does Mathematica have a function to find all singularities of an expression? [duplicate]

Question

This question already has an answer here:

• Is there a built-in function which detects singularities in a function? 2 answers

I am looking for function in Mathematica, which finds all singularities in an expression. To keep it simple, the variable is only one, say x.

So given expression 1/x, it will return $\infty$ and given x*y + 1/(x*y) it will return $x=0,x=\infty,x=-\infty$. Similar to Maple's singular function. "The singular function will return non-removable as well as removable singularities"

Here are some examples:

     singular((c - (a + b + 1)*x)/(x*(1 - x)),x);
          {x = 0}, {x = 1}

and

     singular(exp(1/x),x);
         {x = 0}

and

    singular(2*x/((x-1)*(2*x-1)),x);
          {x = 1}, {x = 1/2}

The expression will always have the variable $x$ in it. It can be rational or not, and it can be basically any valid expression in x. If no singularities exist, then it returns nothing.

I looked, and not able to find this function in Mathematica.

The question is: What would be the closest thing in Mathematica to the above Maple function? This is not exactly like finding poles, since the expression does not have to be rational polynomials. I know any implementation of this function will end up using Solve at the end?

Answer 1

Singularities will occur outside the domain of the function. You could use FunctionDomain to get the domain and then negate it (Not).

With

singularityDomain[f_, x_Symbol] := Reduce[! FunctionDomain[f, x]]

Then

singularityDomain[(c - (a + b + 1)*x)/(x*(1 - x)), x]

x == 1 || x == 0

singularityDomain[Exp[1/x], x]

x == 0

singularityDomain[2*x/((x - 1)*(2*x - 1)), x]

x == 1/2 || x == 1

singularityDomain[Log[x]/(x^2 - 1), x]

x == 1 || x <= 0

It is easy to select the list of isolated singularities from here.

Hope this helps.