return the poles and essential singularities of a given mathematical function

Parameters

 singularities - literal name; 'singularities' math_function - Maple name of mathematical function

Description

 • 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}{\mathrm{sin}}{}\left({z}\right)& {z}{\ne }{2}\\ {0}& {z}{=}{2}\end{array}\right\$ (1)
 > f2(z) = 1/(z-3);
 ${\mathrm{f2}}{}\left({z}\right){=}\frac{{1}}{{z}{-}{3}}$ (2)
 > f3(z) = exp(1/z);
 ${\mathrm{f3}}{}\left({z}\right){=}{{ⅇ}}^{\frac{{1}}{{z}}}$ (3)
 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 $z=0$.

Examples

 > $\mathrm{FunctionAdvisor}\left(\mathrm{singularities},\mathrm{arcsin}\right)$
 $\left[{\mathrm{arcsin}}{}\left({z}\right){,}{"No isolated singularities"}\right]$ (4)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{branch_points},\mathrm{arcsin}\right)$
 $\left[{\mathrm{arcsin}}{}\left({z}\right){,}{z}{\in }\left[{-1}{,}{1}{,}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}\right]\right]$ (5)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{branch_points},\mathrm{exp}\right)$
 $\left[{{ⅇ}}^{{z}}{,}{"No branch points"}\right]$ (6)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{singularities},\mathrm{exp}\right)$
 $\left[{{ⅇ}}^{{z}}{,}{z}{=}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}\right]$ (7)

The value of the function at its singularities can typically be checked by direct evaluation or using eval.

 > $\mathrm{exp}\left(\mathrm{\infty }+I\mathrm{\infty }\right)$
 ${\mathrm{undefined}}{+}{\mathrm{undefined}}{}{I}$ (8)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{singularities},\mathrm{arccot}\right)$
 $\left[{\mathrm{arccot}}{}\left({z}\right){,}{z}{=}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}\right]$ (9)
 > $\mathrm{eval}\left(\mathrm{arccot}\left(z\right),z=\mathrm{\infty }+\mathrm{\infty }I\right)$
 ${\mathrm{undefined}}$ (10)

References

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