Integration of Functions of a Complex Variable

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

Chapter 4 in Complex Analysis, 2023, pp 81-105 from Springer

Abstract: Abstract In the previous two chapters, it was shown that analytic complex-valued functions enjoy excellent differentiability properties that their real counterparts do not share. It is well known that differentiation and integration are mutually inverse operations and they are the main concerns of calculus. To continue on, the next logical step is to consider the integration in the complex plane, as initiated by the French mathematician Augustin-Louis Cauchy (1789–1857). Integration is impossible without the concept of an antiderivative, which becomes much more complicated in complex analysis. For example, it turns out that there are analytic functions in some domains that have no antiderivatives. In this chapter, we will introduce a new concept of an antiderivative along a curve and study its properties. We will also show that the beauty of complex integration also goes far beyond real analysis and prove very important and interesting theorems.

Date: 2023
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DOI: 10.1007/978-3-031-39615-1_4

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