--- name: math-thumb-complex-analysis description: "Apply Chapter 6 (Complex Analysis: Turn Analytic Structure into Bounds and Counts) of Mathematical Rules of Thumb to solve, check, or teach problems. Use it to use complex structure to obtain a valid bound or zero count before seeking exact values." --- # Complex Analysis: Turn Analytic Structure into Bounds and Counts Use this chapter to help the reader make a checked mathematical decision. All 10 numbered rules are available in [the chapter source](references/chapter.md). [The workbook](references/notebook.md) contains a lab, exercises, solutions, and the full rule checklist. [The local rule index](references/rules.json) supplies discovery metadata. ## Start from the reader's task Infer solve, learn, or audit mode from the request. In solve mode, use their supplied numbers and target; in learn mode, use the workbook or their chosen rule; in audit mode, inspect their actual calculation before replacing it. Gather only missing information that changes the choice: Complex function and analytic domain; contour and orientation; singularities; branch choices; target bound or count. If the question falls outside this chapter, say which mathematical operation is missing and suggest a relevant chapter. If the whole-book skill is available, it can carry the task onward, but this chapter works independently. ## Select and apply a rule - **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. Read the selected complete profile, including its equation, “How to read it,” and “How to use it.” The compact graph assumptions are search cues, not a substitute for the profile. Preserve the numbered citation and role. **Independent**, **Workflow**, and **Specialized** describe the relationship to a calculation; exactness, approximation, bound, diagnostic, and heuristic describe a different dimension. Use verified inputs, show the substitution and units, and interpret the result in the reader's decision. Verify by an appropriate bound, alternative computation, limiting case, residual with conditioning, or sensitivity check. If a required condition fails, reject that use and give the specific missing information or alternative method; do not calculate a plausible-looking answer from an invalid formula. **Essential boundary:** Do not infer analyticity from Cauchy-Riemann equations at one point, and do not ignore poles, branch cuts, boundary zeros, or contour orientation. For a sufficient independent result, stop with the decision it supports. For a workflow or specialized rule, name the downstream calculation still needed. A numerical demonstration is evidence for that instance, not a universal proof. ## Teach and check understanding Use the [workbook](references/notebook.md) for guided practice. Start with 6.2.1, 6.2.2, 6.3.3 when the reader wants a starting exercise. Ask for an attempt, offer a relevant hint, and reveal the answer when requested or when teaching requires it. Do not force a quiz when the reader asked for a worked solution. Check whether the reader can explain the controlling quantity, apply the rule to a changed input, identify an invalid use, and distinguish a final answer from a preparatory step. Track only demonstrated work. Give a short prerequisite explanation when needed; avoid requiring completion of earlier chapters. ## Return a usable result Include the chosen rule numbers, assumptions that matter, calculation, verification, and next action. For ongoing work, offer this compact record: question; inputs and units; rules; claim type; book role; assumption status; result and error; check; decision; unresolved next step. Write a progress file only when asked or within an already authorized notebook-editing task. The source chapter is a fixed book snapshot. Preserve its mathematical qualifications and historical evidence gaps. Use outside material only when the reader's task needs it, verify material facts appropriately, and identify that material separately from the book. ## Illustrated exploration Open [the browser reader](assets/reader.html) or [the saved illustrated notebook](assets/notebook.ipynb). [Equation cards](references/equations.json) record the book rule, formula, fixed inputs, supported choices, assumptions and executed default results. - **C06-D01: See powers multiply complex arguments**: rule 6.1.1. - **C06-D02: Use an analytic disk to bound derivatives**: rule 6.2.2. - **C06-D03: Check strict boundary domination before counting zeros**: rule 6.3.3. - **C06-D04: Count only the poles the contour encloses**: rule 6.3.2. Use a saved illustration only when its conditions fit. Browser controls select finite precomputed choices; they do not calculate arbitrary reader inputs. For different inputs, make a checked calculation using the selected rule. Explain what changes, and never claim the notebook ran or the browser was viewed unless it did. Offer prediction questions for learning; answer direct requests without a mandatory quiz.