非标准复分析学习资源推荐问询

Hey there! Glad to hear you're diving into nonstandard analysis—Kelsier's materials are a solid starting point. When it comes to nonstandard complex analysis, there are a few go-to resources I can point you to once you wrap up those texts:

• Non-standard Analysis by Abraham Robinson: The original foundational work in the field. It includes dedicated sections on complex analysis, covering nonstandard characterizations of holomorphic functions (using infinitesimal neighborhoods to define differentiability), nonstandard proofs of core results like the Cauchy Integral Theorem, and extensions to topics like analytic continuation. It’s a bit dense, but essential for understanding the roots of nonstandard complex analysis.
• Lectures on the Hyperreals by Robert Goldblatt: This book balances accessibility with rigor, and its later chapters focus explicitly on nonstandard complex analysis. It connects nonstandard concepts (like hyperfinite approximations, infinitesimal continuity) to standard complex analysis topics, making it a great bridge from your current foundational studies to specialized complex content.
• Introduction to the Theory of Infinitesimals by Stroyan and Luxemburg: A comprehensive text that includes detailed nonstandard treatments of complex analysis, particularly around topics like residue theory, holomorphic function extensions, and the nonstandard perspective on compactness in the complex plane. It’s perfect if you want to dig into the more technical theoretical details.
• An Introduction to Nonstandard Real Analysis by Hurd and Loeb: While the title emphasizes real analysis, it has appendices and scattered sections that extend nonstandard methods to complex analysis—especially useful if you’re interested in intersections with measure theory on the complex plane later on.

Once you’re comfortable with these texts, checking out specialized research papers on nonstandard complex analysis (focused on areas like holomorphic dynamics or complex manifolds) can help you apply these ideas to more advanced topics.

备注：内容来源于stack exchange，提问作者Daniel Schwartz