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 $f\left(z\right)$ at $\mathrm{z0}$ is isolated when $f\left(z\right)$ is discontinuous at $\mathrm{z0}$ but it is analytic in the neighborhood of $\mathrm{z0}$. 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 $f\left(z\right)$ at $\mathrm{z0}$ is removable when there exists a function $g\left(z\right)$ such that $f\left(z\right)=g\left(z\right)$ for $z\ne \mathrm{z0}$ and $g\left(z\right)$ is analytic at $\mathrm{z0}$. The singularity is a pole when $f\left(z\right)=\frac{A\left(z\right)}{B\left(z\right)}$ and both $A\left(z\right),B\left(z\right)$ are analytic at $\mathrm{z0}$ and $A\left(\mathrm{z0}\right)\ne 0,B\left(\mathrm{z0}\right)=0$. 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);

$\mathrm{f1}\left(z\right)=\left\{\begin{array}{cc}\sin \left(z\right)& z\ne 2\\ 0& z=2\end{array}\right.$

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

$\mathrm{f2}\left(z\right)=\frac{1}{z-3}$

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

$\mathrm{f3}\left(z\right)={\&ExponentialE;}^{\frac{1}{z}}$

where $\mathrm{f1}\left(z\right)$ has a removable singularity at $z=2$, $\mathrm{f2}\left(z\right)$ has a pole $z=3$, and $\mathrm{f3}\left(z\right)$ has an essential singularity at $z=0$.

$\mathrm{FunctionAdvisor}\left(\mathrm{singularities}\&comma;\arcsin \right)$

$\left[\arcsin \left(z\right)\&comma;''No\; isolated\; singularities''\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_points}\&comma;\arcsin \right)$

$\left[\arcsin \left(z\right)\&comma;z\in \left[\mathrm{-1}\&comma;1\&comma;\mathrm{\infty }+\mathrm{\infty }\mathrm{I}\right]\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{branch\_points}\&comma;\exp \right)$

$\left[{\&ExponentialE;}^z\&comma;''No\; branch\; points''\right]$

$\mathrm{FunctionAdvisor}\left(\mathrm{singularities}\&comma;\exp \right)$

$\left[{\&ExponentialE;}^z\&comma;z=\mathrm{\infty }+\mathrm{\infty }\mathrm{I}\right]$

$\exp \left(\mathrm{\infty }+\mathrm{I}\mathrm{\infty }\right)$

$\mathrm{undefined}+\mathrm{undefined}\mathrm{I}$

$\mathrm{FunctionAdvisor}\left(\mathrm{singularities}\&comma;\mathrm{arccot}\right)$

$\left[\mathrm{arccot}\left(z\right)\&comma;z=\mathrm{\infty }+\mathrm{\infty }\mathrm{I}\right]$

$\mathrm{eval}\left(\mathrm{arccot}\left(z\right)\&comma;z=\mathrm{\infty }+\mathrm{\infty }\mathrm{I}\right)$

$\mathrm{undefined}$

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