rudin-real-complex-analysis

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?
2. Use Lebesgue dominated convergence
3. Check Fatou's lemma for liminf/limsup
4. Apply Fubini-Tonelli for iterated integrals

1. Complex Analysis Problem?
2. Check analyticity via Cauchy-Riemann
3. For integrals: residue theorem
4. For mappings: Schwarz lemma, conformal properties

1. Functional Analysis?
2. Riesz representation for duals
3. Hahn-Banach for extensions
4. 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.