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Analytic Continuations

In: Complex Analysis

Author

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  • Taras Mel’nyk

    (Taras Shevchenko National University of Kyiv, Faculty of Mathematics and Mechanics)

Abstract

Analytic functions have surprised us in previous chapters with their properties. In this chapter we will learn about another interesting property of analytic functions, namely the possibility of continuing an analytic function from the domain in which it is defined to a larger one. An important property of the analytic continuation procedure is that it is unique. In essence, this means that knowing the values of an analytic function in a small domain, or even on a curve, uniquely determines the value of the function at any other point where that function can be continued. This is a rather surprising and extremely strong assertion. Moreover, for some analytic functions, the analytic continuation can lead to a new concept of a function, which we have already experienced empirically in Sects. 3.6 and 3.7 . Here we will find out under which conditions the analytic continuation leads to a multi-valued function, and under which conditions the newly extended function is single-valued. Along the way we will be introduced to various continuation techniques and other fundamental concepts of complex analysis such as monodromy, global analytic functions, their singularities and Riemann surfaces.

Suggested Citation

  • Taras Mel’nyk, 2023. "Analytic Continuations," Springer Books, in: Complex Analysis, chapter 8, pages 185-215, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-39615-1_8
    DOI: 10.1007/978-3-031-39615-1_8
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