FunctionAdvisor/singularities
return the poles and essential singularities of a given mathematical function
Calling Sequence
Parameters
Description
Examples
References
FunctionAdvisor(singularities, math_function)
singularities
-
literal name; 'singularities'
math_function
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 fz at z0 is isolated when fz 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 fz at z0 is removable when there exists a function gz such that fz=gz for z≠z0 and gz is analytic at z0. The singularity is a pole when fz=AzBz and both Az,Bz are analytic at z0 and Az0≠0,Bz0=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);
f1z=sinzz≠20z=2
f2(z) = 1/(z-3);
f2z=1z−3
f3(z) = exp(1/z);
f3z=ⅇ1z
where f1z has a removable singularity at z=2, f2z has a pole z=3, and f3z has an essential singularity at z=0.
FunctionAdvisorsingularities,arcsin
arcsinz,No isolated singularities
FunctionAdvisorbranch_points,arcsin
arcsinz,z∈−1,1,∞+∞I
FunctionAdvisorbranch_points,exp
ⅇz,No branch points
FunctionAdvisorsingularities,exp
ⅇz,z=∞+∞I
The value of the function at its singularities can typically be checked by direct evaluation or using eval.
exp∞+I∞
undefined+undefinedI
FunctionAdvisorsingularities,arccot
arccotz,z=∞+∞I
evalarccotz,z=∞+∞I
undefined
Brown, J.W. and Churchill, R.V. Complex Variables and Applications. 6th Ed. McGraw-Hill Science/Engineering/Math, 1995.
See Also
DEtools[singularities]
eval
FunctionAdvisor
FunctionAdvisor/branch_cuts
FunctionAdvisor/branch_points
FunctionAdvisor/topics
singular
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