return the poles and essential singularities of a given mathematical function
literal name; 'singularities'
Maple name of mathematical function
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⁡z at z0 is isolated when f⁡z is discontinuous at z0 but it is analytic in the neighborhood of 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⁡z at z0 is removable when there exists a function g⁡z such that f⁡z=g⁡z for z≠z0 and g⁡z is analytic at z0. The singularity is a pole when f⁡z=A⁡zB⁡z and both A⁡z,B⁡z are analytic at z0 and A⁡z0≠0,B⁡z0=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);
f2(z) = 1/(z-3);
f3(z) = exp(1/z);
where f1⁡z has a removable singularity at z=2, f2⁡z has a pole z=3, and f3⁡z has an essential singularity at z=0.
arcsin⁡z,No isolated singularities
ⅇz,No branch points
The value of the function at its singularities can typically be checked by direct evaluation or using eval.
Brown, J.W. and Churchill, R.V. Complex Variables and Applications. 6th Ed. McGraw-Hill Science/Engineering/Math, 1995.
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