Understanding Poles and Zeros in Complex Analysis

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SUMMARY

The function 1/(z-w)^4 has one pole of order 4 located at z=w. To determine the zeros of a function, one must set the numerator equal to zero, while poles are identified by setting the denominator to zero. The term "singularity" encompasses various types, including removable, pole, and essential singularities, with poles being a specific category of singularity.

PREREQUISITES
  • Understanding of complex functions
  • Knowledge of poles and zeros in complex analysis
  • Familiarity with the concept of singularities
  • Basic algebraic manipulation of functions
NEXT STEPS
  • Study the classification of singularities in complex analysis
  • Learn about the residue theorem and its applications
  • Explore the concept of analytic functions and their properties
  • Investigate the implications of pole order on function behavior
USEFUL FOR

Students and professionals in mathematics, particularly those focusing on complex analysis, as well as engineers and physicists dealing with complex functions in their work.

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Oh god, so confused and panicked today:cry:
I know this is a very basic question, but, givin the function 1/(z-w)^4

does this have one pole of order 4, or possibly 4 poles of order 1...?

Also, could you please clarify,
''to get the zero's of a function, set the numerator = 0''
''to get the poles of a function, set the denominator = 0''
is this correct??
 
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(z-w)^4 is 0 iff z=w. So, it has one pole at w.

I always though when the denominator was 0 it was referred to as a singularity?
 
as yes, right you are, and a pole is one of the three types of singularity... removable, pole and essential!
 

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