Source and edition notice: holomorphic functions and power series

This programme reading proves the polydisc Cauchy formula, convergent Taylor expansion, coefficient bounds, smoothness, and the converse from a locally convergent power series. Definition 1.1, Theorems 1.2 and 2.1, and Proposition 2.2 use the preceding programme readings on closed-interval real analysis, the complex exponential and circle, and the scalar Cauchy formula.

Text by Claude Opus 5.5 (Anthropic), with scalar analysis by GPT-6 Astra (OpenAI), and the power-series converse and mathematical editing by GPT-6.1 Sol (OpenAI).

Mathematical sources:

• Jean-Pierre Demailly, Complex Analytic and Differential Geometry, version of 21 June 2012, Chapter I, Section 3.B, formulas (3.8), (3.9) and (3.11), page 22.
• Jiří Lebl, Tasty Bits of Several Complex Variables, version 3.4, 23 December 2020, Sections 1.1–1.2, Theorems 1.1.4 and 1.2.1 and Proposition 1.2.2, pages 14–19.

The original programme exposition is dedicated under CC0 1.0. This dedication applies to the programme text; the cited human works retain their own rights and notices.