rudin-real-complex-analysis
Scannednpx machina-cli add skill parcadei/Continuous-Claude-v3/rudin-real-complex-analysis --openclawRudin'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.
Source
git clone https://github.com/parcadei/Continuous-Claude-v3/blob/main/.claude/skills/math/rudin-real-complex-analysis/SKILL.mdView on GitHub Overview
This skill is a Rudin Real and Complex Analysis reference for graduate-level problem solving. It centers on measure theory, Lebesgue integration, Lp spaces, functional analysis, and complex analysis, with a focus on applying standard techniques and connecting the real and complex viewpoints.
How This Skill Works
Use a three-way decision tree to classify problems as Measure/Integration, Complex Analysis, or Functional Analysis. Then apply Rudin-aligned techniques: Dominated Convergence, Fatou's lemma, and Fubini-Tonelli for real analysis; Cauchy-Riemann, contour integrals and residues for complex analysis; and Riesz representation, Hahn-Banach, open/closed mapping theorems for functional analysis. The skill provides example commands and references to key theorems to guide solution structure.
When to Use It
- Solving Lebesgue integration problems and interchange of limits
- Analyzing Lp spaces and duals in functional analysis
- Working with analytic functions, contour integration, and residues in complex analysis
- Exploring connections between real and complex analysis within Rudin's framework
- Applying foundational theorems like Dominated Convergence, Cauchy-Riemann, and the Residue Theorem
Quick Start
- Step 1: Classify the problem as Real/Complex/Functional analysis
- Step 2: Apply the corresponding theorem set (DCT, C-R, Riesz/Hahn-Banach, etc.)
- Step 3: Translate the result back into Rudin's framework and verify conditions
Best Practices
- Identify the problem domain (real, complex, or functional) using the decision tree
- Map to the applicable theorems early (e.g., Dominated Convergence, Cauchy-Riemann, Hahn-Banach)
- Check hypotheses (integrability, analyticity, boundedness) before applying theorems
- Cross-verify results with both real and complex perspectives when possible
- Annotate steps with Rudin's context and relevant chapter references
Example Use Cases
- Prove interchange of limit and integral using Dominated Convergence (Lebesgue context)
- Compute a real integral via the Residue Theorem from complex analysis
- Show a function is holomorphic by verifying Cauchy-Riemann equations
- Identify dual space via the Riesz Representation Theorem
- Demonstrate a conformal mapping between domains
