singularities - Maple Help
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FunctionAdvisor/singularities

return the poles and essential singularities of a given mathematical function

 

Calling Sequence

Parameters

Description

Examples

References

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  at  is isolated when  is discontinuous at  but it is analytic in the neighborhood of . 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  at  is removable when there exists a function  such that  for  and  is analytic at . The singularity is a pole when  and both  are analytic at  and . 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);

(1)

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

(2)

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

(3)
  

where  has a removable singularity at ,  has a pole , and  has an essential singularity at .

Examples

(4)

(5)

(6)

(7)

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

(8)

(9)

(10)

References

  

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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