return the poles and essential singularities of a given mathematical function - Maple Help

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FunctionAdvisor/singularities - return the poles and essential singularities of a given mathematical function

Calling Sequence

FunctionAdvisor(singularities, math_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 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 zz0 and gz is analytic at z0. The singularity is a pole when fz=AzBz and both Az,Bz are analytic at z0 and Az00,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&equals;&lcub;sinzz20z&equals;2

(1)

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

f2z&equals;1z3

(2)

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

f3z&equals;&ExponentialE;1z

(3)
  

where f1z has a removable singularity at z&equals;2, f2z has a pole z&equals;3, and f3z has an essential singularity at z&equals;0.

Examples

FunctionAdvisorsingularities&comma;arcsin

arcsinz&comma;No isolated singularities

(4)

FunctionAdvisorbranch_points&comma;arcsin

arcsinz&comma;z&in;1&comma;1&comma;&infin;&plus;&infin;I

(5)

FunctionAdvisorbranch_points&comma;exp

&ExponentialE;z&comma;No branch points

(6)

FunctionAdvisorsingularities&comma;exp

&ExponentialE;z&comma;z&equals;&infin;&plus;&infin;I

(7)

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

&ExponentialE;&infin;&plus;I&infin;

undefined&plus;undefinedI

(8)

FunctionAdvisorsingularities&comma;arccot

arccotz&comma;z&equals;&infin;&plus;&infin;I

(9)

arccotzz&equals;&infin;&plus;&infin;I|arccotzz&equals;&infin;&plus;&infin;I

undefined

(10)

See Also

DEtools[singularities], eval, FunctionAdvisor, FunctionAdvisor/branch_cuts, FunctionAdvisor/branch_points, FunctionAdvisor/topics, singular

References

  

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


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