# Chapter 6 notebook: Complex Analysis: Turn Analytic Structure into Bounds and Counts **Goal:** Use complex structure to obtain a valid bound or zero count before seeking exact values. **Start with:** Rules 6.2.1, 6.2.2, 6.3.3. Use a calculator or the optional Python lab. Programming is optional for the workbook; the Jupyter version requires a Python 3 kernel. All lab numbers are constructed practice inputs, not observed data. **Source:** [Finished Chapter 6](../skills/math-thumb-complex-analysis/references/chapter.md) from *Mathematical Rules of Thumb*. Numbered rules, classifications, and the decision path come from the book. The lab, exercises, answer key, and worksheet prompts are companion additions. ## 1. Frame your decision Write a question in which an answer would change something you do. Gather: Complex function and analytic domain; contour and orientation; singularities; branch choices; target bound or count. - My question and intended decision: - Known inputs and units: - Required accuracy or threshold: - What I expect before calculating: - What I still need to find out: Use the lab as a worked starting point if you do not yet have your own problem. For notation or prerequisites, ask the chapter skill to explain only the concept blocking the next step. ## 2. Choose a route - **Is the main operation multiplication or exponentiation?** Move to polar form. If a logarithm, root, or fractional power appears, state the branch before simplifying. - **Are you about to invoke an analytic theorem?** Write $f=u+iv$, screen with Cauchy–Riemann, and verify the required neighborhood and regularity. - **Do you need a bound rather than an exact integral?** Estimate maximum integrand magnitude and contour length. Split the path if one maximum is too crude. - **Do you know a boundary magnitude on an analytic disk?** Use Cauchy's estimate for derivatives; use the maximum-modulus principle for the function's largest modulus. - **Does a quotient have simple poles?** Compute each needed residue by numerator value over denominator derivative. - **Is the contour integral closed?** List and classify enclosed poles, check orientation, and use the residue theorem. - **Is the real question a zero count?** Try Rouché when one term dominates on the boundary. Use the argument principle when $f'/f$ or boundary winding is more accessible. **Chapter-specific stop check:** Do not infer analyticity from Cauchy-Riemann equations at one point, and do not ignore poles, branch cuts, boundary zeros, or contour orientation. ## 3. Work the lab Suppose f is analytic on a neighborhood of the closed disk |z|<=2 and |f(z)|<=3 on its boundary. Cauchy's estimate gives |f''(0)| <= 2! × 3/2^2=1.5. The same boundary data give a contour-integral magnitude bound M × L=3 × (2 × pi × 2)=12 × pi. For the integral of f itself, analyticity actually gives zero, showing how a valid bound can still be loose. Predict the sign and scale before running the code. Then change one input and explain why the result moves. The code checks the constructed example; it does not prove the rule for every possible input. Code assertions may describe the example's chosen regime, so inspect them before changing that regime. ```python import math radius, maximum, derivative_order = 2.0, 3.0, 2 assert radius > 0 and maximum >= 0 and derivative_order >= 0 derivative_bound = math.factorial(derivative_order)*maximum/radius**derivative_order ml_bound = maximum*2*math.pi*radius print(f"Derivative bound: {derivative_bound:.6f}; ML bound: {ml_bound:.6f}") ``` **My prediction, observed result, and explanation:** _Record your work here._ ## 4. Practise without the answers ### Exercise 1 Count zeros of z^3+0.1 inside |z|=1 using Rouche. **My approach, assumptions, calculation, and check:** _Write your attempt here._ ### Exercise 2 Evaluate the counterclockwise contour integral of 1/(z-0.5) around |z|=1. **My approach, assumptions, calculation, and check:** _Write your attempt here._ ### Exercise 3 Why can the disk maximum-modulus argument not be applied to 1/z on |z|<=1? **My approach, assumptions, calculation, and check:** _Write your attempt here._ **Coaching prompt:** “Use the Chapter 6 skill to help me with Exercise 2. Ask for my attempt, give one useful hint if I need it, and help me check the assumptions before showing the answer.” ## 5. Answer key and reasoning Read this after attempting the exercises, or use it immediately if you prefer a complete walkthrough. An answer is complete only when its assumptions and stopping point are clear. ### Answer 1 On the boundary, |0.1|<|z^3|=1. Both terms are analytic, so the polynomial has the same number of zeros as z^3: three, counting multiplicity. ### Answer 2 There is one enclosed simple pole with residue 1; the integral is 2 × pi × i. Clockwise orientation would reverse its sign. ### Answer 3 A pole at zero violates analyticity on the disk. Its boundary modulus is one, but its interior magnitude is unbounded near zero. ## 6. Build the complete chapter toolkit The new lab samples the chapter; the following checklist covers all 10 rules. Study one thematic group at a time. A large group can take several sessions. - **Choose a Representation That Exposes Structure:** work with rules 6.1.1, 6.1.2, 6.1.3. - **Convert Analyticity into Immediate Bounds:** work with rules 6.2.1, 6.2.2, 6.2.3. - **Turn Singularities and Boundary Behavior into Counts:** work with rules 6.3.1, 6.3.2, 6.3.3, 6.3.4. For each selected rule, read its equation, explanation, and worked use in the source. Reproduce that example; change one input; then change one assumption so the rule is no longer justified. Record the result and what check catches the failure. Historical examples remain labeled and qualified as in the source. Read each complete numbered profile in the [chapter reference](../skills/math-thumb-complex-analysis/references/chapter.md) before applying it. The cues below abbreviate the graph metadata; they are not complete conditions. Change the status only after doing the practice described below. | Rule | Book role | First assumptions to inspect | Practice status | |---|---|---|---| | 6.1.1: Use polar form for complex products and powers | Independent | consistent angle units | new | | 6.1.2: Treat the principal argument as branch-dependent | Workflow | declared branch interval; path avoids branch cut when continuity needed | new | | 6.1.3: Use Cauchy-Riemann equations as a quick analyticity screen | Workflow | real partial derivatives exist for test | new | | 6.2.1: Bound a contour integral by maximum times length | Independent | rectifiable contour; valid uniform modulus bound | new | | 6.2.2: Use Cauchy's estimate to bound derivative scale | Independent | analytic inside and on circle; known boundary modulus bound | new | | 6.2.3: Look on the boundary for analytic maxima | Independent | analytic on connected interior; continuous on compact closure | new | | 6.3.1: Compute a simple-pole residue by division of derivatives | Workflow | isolated simple pole | new | | 6.3.2: Turn a contour integral into a sum of enclosed residues | Workflow | meromorphic inside contour; closed positively oriented contour | new | | 6.3.3: Use Rouché when one boundary term dominates | Independent | analytic functions inside contour; strict boundary dominance | new | | 6.3.4: Count zeros minus poles with the argument principle | Workflow | meromorphic function; no zeros or poles on contour | new | For a completed row, record: **rule number / my new input / mathematical claim type / verified assumptions / calculation / check / valid use / rejected use / next step**. “Practised” means you worked an example. “Demonstrated” means you can explain a valid use, transfer it, and reject a misuse without the answer key. ## 7. Apply it to your own problem Return to your opening question. Choose the smallest rule set that can settle it. Use the book's independent/workflow/specialized classification separately from the claim type (exact, approximate, bound, diagnostic, or heuristic). - Selected rule number(s) and reason: - Assumptions that hold, fail, or remain uncertain: - Substitution with units: - Result and error, uncertainty, or bound: - Independent check or limiting case: - Decision this supports: - Stop here, do a named next calculation, or gather missing information: ## 8. Transfer and continue Useful nearby chapters: Chapter 5: analysis; Chapter 11: asymptotics; Chapter 24: signal processing. Bring the question, units, assumptions, result type, and uncertainty to the next chapter. Choose a bridge only when it supplies an operation you actually need. **Completion check:** Explain this chapter's lab in your own words; solve one changed-input exercise; reject one invalid use; and produce a decision record for your own problem. If one check fails, revisit that part of the chapter rather than marking every rule complete.