add(analysis-001): cosine power–cosine Fourier integral

[izolyte]

What & why

Adds the cosine power–cosine Fourier integral as the first analysis problem.
The result — a closed form via Gamma functions — is a classical identity that
warrants two independent proofs at different levels of machinery.

Closes #9

Checklist

• Compiles: typst compile --root . problems/analysis-001-cos-power-cos-integral.typ — clean
• Index in sync: ran bash scripts/build-index.sh — 2 problems
• Meets the authoring standard in docs/AUTHORING.md
• Every result used-but-not-proved carries a cite(...) — no red [ref?] markers
• Source set in metadata: Romanian Mathematical Magazine, K. Srinivasa Raghava

Bilingual

• Thai sibling analysis-001-cos-power-cos-integral-th.typ — both compile

Notes

Two proofs, one = Proofs section with == subsections (new convention for multi-proof problems, documented in write-proof skill):

• Proof I — strong induction on $n$: IBP → product-to-sum recursion → Carlson's theorem extends to real $\alpha$
• Proof II — complex exponential: binomial expansion → Vandermonde–Chu identity → Euler reflection formula

Unit circle diagram included for Proof II showing the $w = e^{-2ix}$ contour path.
Recommended review order: Problem Statement → Proof I → Proof II → Verification.

Summary by CodeRabbit

• New Features
  • Added a new solved problem: “Cosine Power–Cosine Fourier Integral”, including a full solution document with two distinct proof approaches and verification checks.
• Documentation
  • Updated the README problem index and progress summary to include the new solved entry and expanded the “Topics covered” list.
• Chores
  • Adjusted index generation to derive the displayed problem number from the filename prefix for consistent numbering.

[coderabbitai]

No actionable comments were generated in the recent review. 🎉

ℹ️ Recent review info

⚙️ Run configuration

Configuration used: defaults

Review profile: CHILL

Plan: Pro Plus

Run ID: 522fe5ea-8438-4c34-9111-086f1d776588

📥 Commits

Reviewing files that changed from the base of the PR and between 3b1d837 and f7891a6.

📒 Files selected for processing (4)

• README.md
• problems/analysis-001-cos-power-cos-integral-th.typ
• problems/analysis-001-cos-power-cos-integral.typ
• scripts/build-index.sh

✅ Files skipped from review due to trivial changes (1)

• README.md

🚧 Files skipped from review as they are similar to previous changes (2)

• problems/analysis-001-cos-power-cos-integral-th.typ
• problems/analysis-001-cos-power-cos-integral.typ

---

📝 Walkthrough

Walkthrough

Two new Typst documents are added for problem analysis-001, presenting the statement and two independent proofs of the closed-form evaluation of ∫₀^{π/2} cosᵅ(x)cos(βx) dx for β > α > −1 in terms of Gamma functions. The README index and progress summary are updated to reflect the new entry, and the build script is updated to extract the full problem identifier from filenames.

Changes

Cosine Power–Cosine Fourier Integral problem solution

Layer / File(s)	Summary
Document setup and problem statement problems/analysis-001-cos-power-cos-integral-th.typ, problems/analysis-001-cos-power-cos-integral.typ	Both files define Typst metadata, imports, and Cetz package setup; introduce I(α,β) and F(α,β) with Gamma-function properties; include diagrams for the integrand sign structure and the overall proof strategy.
Proof I: strong induction and recurrence problems/analysis-001-cos-power-cos-integral-th.typ, problems/analysis-001-cos-power-cos-integral.typ	Both files derive base cases (β=0 and β≠0), establish an integration-by-parts recurrence for I(n,β) using product-to-sum identities, verify that F(n,β) satisfies the same recurrence, and invoke Carlson's theorem to extend the equality to real α > −1.
Proof II: complex-exponential and binomial expansion problems/analysis-001-cos-power-cos-integral-th.typ, problems/analysis-001-cos-power-cos-integral.typ	Both files rewrite cos(βx) as Re(e^{iβx}), apply the binomial theorem to cosⁿ(x), evaluate the resulting exponential integrals, collapse the alternating binomial sum via Vandermonde–Chu, apply Euler's reflection formula to match F(n,β), and extend to real α > −1 via Carlson's theorem.
Verification, observations, and index update problems/analysis-001-cos-power-cos-integral-th.typ, problems/analysis-001-cos-power-cos-integral.typ, scripts/build-index.sh, README.md	Both files compute explicit checks for sample parameter pairs and the Wallis special case, note integer-reduction and convergence properties, and give a Beta-function reformulation; the build script is updated to extract the full problem identifier, and the README adds the new problem row and updates the solved count and topics list.

Estimated code review effort

🎯 3 (Moderate) | ⏱️ ~25 minutes

Poem

A rabbit with chalk in paw,
Scribbled integrals wall to wall,
cos^α times cos(βx),
Gamma functions answered the call! 🐇✨
Two proofs drawn, Carlson extends—
A Fourier tale that never ends! 🎉

🚥 Pre-merge checks | ✅ 5

✅ Passed checks (5 passed)

Check name	Status	Explanation
Description Check	✅ Passed	Check skipped - CodeRabbit’s high-level summary is enabled.
Title check	✅ Passed	Title accurately summarizes the main change: adding analysis-001, a new problem on the cosine power-cosine Fourier integral, which is the primary focus of the PR.
Linked Issues check	✅ Passed	The PR implements the core requirement from issue #9: proving the closed-form integral formula using Gamma functions with two independent proofs, bilingual support, and proper source attribution.
Out of Scope Changes check	✅ Passed	All changes are scoped to the analysis-001 problem: the new problem files, README updates reflecting the new entry, and a script modification to properly index the new file naming convention.
Docstring Coverage	✅ Passed	No functions found in the changed files to evaluate docstring coverage. Skipping docstring coverage check.

_{✏️ Tip: You can configure your own custom pre-merge checks in the settings.}

✨ Finishing Touches

📝 Generate docstrings

• Create stacked PR
• Commit on current branch

🧪 Generate unit tests (beta)

• Create PR with unit tests
• Commit unit tests in branch problem/analysis-001-cos-power-cos-integral

---

_{Comment @coderabbitai help to get the list of available commands and usage tips.}

[coderabbitai]

Actionable comments posted: 3

🤖 Prompt for all review comments with AI agents

Verify each finding against current code. Fix only still-valid issues, skip the
rest with a brief reason, keep changes minimal, and validate.

Inline comments:
In `@problems/analysis-001-cos-power-cos-integral-th.typ`:
- Around line 246-247: The induction claim at lines 246-247 states the identity
holds for all β in ℝ, but the subsequent proof divides by β and by n²-β², and
Proof II requires s=(β-n)/2>0, which means β>n. Replace the domain "สำหรับทุก
$gamma in RR$" (for all gamma in ℝ) with a restricted domain statement such as
"สำหรับทุก $gamma > n$" to match the actual derivation constraints.
Alternatively, if the identity should hold on the full real line, add a separate
continuity or limit argument that extends the result from the restricted domain
β>n to the excluded values where division is undefined or the constraint is
violated.

In `@problems/analysis-001-cos-power-cos-integral.typ`:
- Around line 256-257: The inductive hypothesis statement around line 256-257
claims validity for all real β, but the proof contains divisions by β and n±β,
and applies formulas with domain restrictions like s=(β-n)/2>0. Revise the
statement of the inductive hypothesis to specify that I(n-1, β) = F(n-1, β) only
for β in the valid domain (β > α > -1) as stated in the theorem, rather than
claiming it holds for all real β. Alternatively, explicitly separate and handle
the singular or boundary cases where these divisions are undefined or the
formulas do not apply. Apply this same correction to the analogous statements
around lines 393-395.

In `@README.md`:
- Around line 41-43: The problem table in README.md has duplicate index numbers
in the # column, with both the "Cosine Power–Cosine Fourier Integral" entry and
the "A Nested Floor–Ceiling Integral" entry labeled as 001. Update the second
entry (A Nested Floor–Ceiling Integral) to have a unique index number of 002 to
ensure each problem has a distinct, sequential identifier in the # column.

🪄 Autofix (Beta)

Fix all unresolved CodeRabbit comments on this PR:

• Push a commit to this branch (recommended)
• Create a new PR with the fixes

---

ℹ️ Review info

⚙️ Run configuration

Configuration used: defaults

Review profile: CHILL

Plan: Pro Plus

Run ID: 4b2ab967-2a03-4844-bb20-89b2da9ec9ba

📥 Commits

Reviewing files that changed from the base of the PR and between 0aac680 and 15a9d2f.

📒 Files selected for processing (3)

• README.md
• problems/analysis-001-cos-power-cos-integral-th.typ
• problems/analysis-001-cos-power-cos-integral.typ