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5 - Taylor and Laurent Series

Published online by Cambridge University Press:  05 June 2012

Yue Kuen Kwok
Affiliation:
Hong Kong University of Science and Technology
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Summary

A power series with non-negative power terms is called a Taylor series. In complex variable theory, it is common to work with power series with both positive and negative power terms. This type of power series is called a Laurent series. The primary goal of this chapter is to establish the relation between convergent power series and analytic functions. More precisely, we try to understand how the region of convergence of a Taylor series or a Laurent series is related to the domain of analyticity of an analytic function. The knowledge of Taylor and Laurent series expansion is linked with more advanced topics, like the classification of singularities of complex functions, residue calculus, analytic continuation, etc.

This chapter starts with the definitions of convergence of complex sequences and series. Many of the definitions and theorems for complex sequences and series are inferred from their counterparts in real variable calculus.

Complex sequences and series

An infinite sequence of complex numbers, denoted by {zn}, can be considered as a function defined on a set of positive integers into the unextended complex plane. In other words, the sequence of complex numbers z1, z2, z3, … is arranged sequentially and defined by some specific rule.

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Publisher: Cambridge University Press
Print publication year: 2010

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  • Taylor and Laurent Series
  • Yue Kuen Kwok, Hong Kong University of Science and Technology
  • Book: Applied Complex Variables for Scientists and Engineers
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511844690.006
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  • Taylor and Laurent Series
  • Yue Kuen Kwok, Hong Kong University of Science and Technology
  • Book: Applied Complex Variables for Scientists and Engineers
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511844690.006
Available formats
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To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Taylor and Laurent Series
  • Yue Kuen Kwok, Hong Kong University of Science and Technology
  • Book: Applied Complex Variables for Scientists and Engineers
  • Online publication: 05 June 2012
  • Chapter DOI: https://doi.org/10.1017/CBO9780511844690.006
Available formats
×