Elementary Analytic Functions

Taras Mel’nyk
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Taras Mel’nyk: Taras Shevchenko National University of Kyiv, Faculty of Mathematics and Mechanics

Chapter 3 in Complex Analysis, 2023, pp 43-79 from Springer

Abstract: Abstract Conformal mappings are of immense importance in various branches of mathematics and in many applications. To solve many problems, one needs to be able to construct a bijective conformal mapping from one domain onto another in the complex plane. In this chapter we study how to construct such bijective conformal mappings. We will consider various elementary analytic functions, find domains of univalence and images of these domains. In addition, for many elementary analytic functions in ℂ $$\mathbb C$$ we find their inverses, which in some cases turn out to be multivalued. We introduce the first (intuitive) concept of a Riemann surface for multivalued functions and show how to construct Riemann surfaces for the inverses of elementary analytic functions. As a result of these studies, we will establish facts that are incorrect in real analysis, for example, we can calculate the logarithm of negative numbers and solve the equation sin z = 2. $$\displaystyle \sin z = 2 . $$

Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-39615-1_3

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DOI: 10.1007/978-3-031-39615-1_3

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