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?** - 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 ```bash uv run python scripts/ragie_query.py --query "YOUR_TOPIC measure integration" --partition math-textbooks --top-k 5 ``` ### SymPy for Symbolic Computation ```bash uv run python scripts/sympy_compute.py integrate "exp(-x**2)" --var x --bounds "0,oo" ``` ### Z3 for Verification ```bash 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.