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

1. Measure/Integration Problem?

   • Use Lebesgue dominated convergence
   • Check Fatou's lemma for liminf/limsup
   • Apply Fubini-Tonelli for iterated integrals

2. Complex Analysis Problem?

   • Check analyticity via Cauchy-Riemann
   • For integrals: residue theorem
   • For mappings: Schwarz lemma, conformal properties

3. 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.