rudin-real-complex-analysis

Research

Problem-solving with Rudin's Real and Complex Analysis textbook

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Source SKILL.md: https://github.com/vibeeval/vibecosystem/blob/HEAD/skills/math/rudin-real-complex-analysis/SKILL.md

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Rudin's Real and Complex Analysis

Reference skill for Walter Rudin's "Real and Complex Analysis" (3rd Edition) - a graduate-level text covering measure theory, integration, functional analysis, and complex analysis.

When to Use

Use this skill when working on:

Measure theory and Lebesgue integration
Lp spaces and functional analysis
Complex analysis (analytic functions, contour integration, residues)
Connections between real and complex analysis

Topics Covered

Real Analysis

Limits and continuity in metric spaces
Convergence of sequences and series
Differentiation and integration techniques
Metric spaces and topology

Complex Analysis

Analytic functions and Cauchy-Riemann equations
Contour integration and Cauchy's theorem
Residue theorem and applications
Conformal mappings
Power series representations

Topology

Topological spaces
Compactness and connectedness
Metric space topology

Algebra

Rings and ideals (in context of function spaces)

Decision Tree

Measure/Integration Problem?
- Use Lebesgue dominated convergence
- Check Fatou's lemma for liminf/limsup
- Apply Fubini-Tonelli for iterated integrals
Complex Analysis Problem?
- Check analyticity via Cauchy-Riemann
- For integrals: residue theorem
- For mappings: Schwarz lemma, conformal properties
Functional Analysis?
- Riesz representation for duals
- Hahn-Banach for extensions
- Open mapping/closed graph theorems

Tool Commands

Query Rudin Content

uv run python scripts/ragie_query.py --query "YOUR_TOPIC measure integration" --partition math-textbooks --top-k 5

SymPy for Symbolic Computation

uv run python scripts/sympy_compute.py integrate "exp(-x**2)" --var x --bounds "0,oo"

Z3 for Verification

uv run python scripts/z3_solve.py prove "forall x, |f(x)| <= M implies bounded"

Key Theorems Reference

Theorem	Chapter	Use Case
Dominated Convergence	Ch 1	Interchange limit and integral
Riesz Representation	Ch 2	Identify dual spaces
Cauchy's Theorem	Ch 10	Contour integrals = 0 for analytic
Residue Theorem	Ch 10	Evaluate real integrals
Open Mapping	Ch 5	Surjective bounded linear maps

Cognitive Tools Reference

See .claude/skills/math-mode/SKILL.md for full tool documentation.