The FunctionAdvisor(singularities, math_function) command returns the isolated poles and essential singularities of the function, if any, or the string "No isolated singularities". If the requested information is not available, it returns NULL.

A singularity of  at  is isolated when  is discontinuous at  but it is analytic in the neighborhood of . To compute the branch points of a mathematical function, that is, the non-isolated singularities related to the multivaluedness of the function, use the FunctionAdvisor(branch_point, math_function) command.

An isolated singularity can be removable, essential, or a pole. In the call FunctionAdvisor(singularities, math_func) only poles and essential singularities are returned.

An isolated singularity of  at  is removable when there exists a function  such that  for  and  is analytic at . The singularity is a pole when  and both  are analytic at  and . The singularity is essential when it is neither removable nor a pole.

The following are examples of these types of isolated singularities

f1(z) = piecewise(z <> 2, sin(z), z = 2, 0);

f2(z) = 1/(z-3);

f3(z) = exp(1/z);

where  has a removable singularity at ,  has a pole , and  has an essential singularity at .

Brown, J.W. and Churchill, R.V. Complex Variables and Applications. 6th Ed. McGraw-Hill Science/Engineering/Math, 1995.