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The investigation of Cauchy integrals on the unit circle begins in this chapter, marking the entry into the inherently complex analysis. The Cauchy integral formula for the unit circle states that functions in the disc algebra on the open unit disc \(\mathbb {D}\) coincide with their Cauchy integral. Since Cauchy integrals are analytic functions, the analyticity of holomorphic functions emerges in a magical way, almost as if by itself. This provides the key to unlock the gate to the fascinating world of holomorphic functions. A first glance into this world is immediately taken.

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Magic Circles: Circular Functions and Local Complex Analysis

  • 章节

  • 发表日期：

    2025年

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  • Jürgen Müller

摘要

The investigation of Cauchy integrals on the unit circle begins in this chapter, marking the entry into the inherently complex analysis. The Cauchy integral formula for the unit circle states that functions in the disc algebra on the open unit disc \(\mathbb {D}\) coincide with their Cauchy integral. Since Cauchy integrals are analytic functions, the analyticity of holomorphic functions emerges in a magical way, almost as if by itself. This provides the key to unlock the gate to the fascinating world of holomorphic functions. A first glance into this world is immediately taken.