 Residue CalculusTaras Mel’nyk
Chapter 7 in Complex Analysis, 2023, pp 151-184 from Springer Abstract: Abstract Just as a person’s character is manifested in extreme situations, so the properties of analytic functions are determined by their behavior in isolated singularities. In this chapter, we will illustrate this claim with examples of integral calculations. It turns out that in order to calculate the integral of an analytic function along a curve, it is necessary to determine some values, called residues, of that function at its singularities. The reader can appreciate both the power and the simplicity of the residue theory developed by Cauchy for calculating complicated integrals, including integrals of real-valued functions. This theory helps to deduce amazing formulas both for the decomposition of a meromorphic function, e.g. cot z , $$\cot z,$$ into an infinite sum of simple fractions that are responsible for its poles, and for the factorization of an entire function, e.g. sin z , $$\sin z,$$ into an infinite product of factors that are responsible for its zeros. The theory also supplies a ready-made framework for counting zeros and poles of a given meromorphic function or zeros of an analytic function, in particular, we prove the argument principle and Rouché’s theorem. Date: 2023 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-39615-1_7 Ordering information: This item can be ordered from DOI: 10.1007/978-3-031-39615-1_7 Access Statistics for this chapter More chapters in Springer Books from Springer |
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