Ahlfors — *Complex Analysis* (Fast Track 1.04) — Audit + Gap Plan

Book: Lars V. Ahlfors, Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable (3rd ed., McGraw-Hill, 1979). ISBN 0-07-000657-1. ~331 pages, 8 chapters, ~280 exercises.

Fast Track entry: 1.04. The canonical graduate complex-analysis text — the single-complex-variable anchor of the FT analysis spine. Already cited by tier_anchors on five existing Codex units (06.01.01-holomorphic-function, 06.01.06-riemann-mapping-theorem, and others — see §2). Paired downstream with FT 1.07 (Donaldson / Forster / Miranda Riemann-surfaces); Ahlfors prepares the plane-and-disc machinery the surfaces books then lift to global geometry.

Purpose: lightweight P1-lite + P2 + P3-lite pass per orchestration protocol. Punch-list of new units + deepenings to reach docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4 threshold. Peer audit to forster-riemann-surfaces.md and donaldson-riemann-surfaces.md (the surfaces side of FT 1.07): Ahlfors owns the classical-plane spine those plans assume present but Codex has barely shipped. The Codex chapter 06.01-complex-analysis/ ships 6 units, 5 of which are v0.5 Strand C/D templated stubs (only 06.01.01 carries genuine prose). Topic-level coverage of Ahlfors's first half is shallow but nominally present; the entire Ahlfors back half — gamma function, zeta function, Weierstrass factorization (only the corollary use in Cousin II shipped), normal families / Montel, Schwarz-Christoffel, Picard's theorems, harmonic-function theory, hyperbolic / Schwarz-Pick geometry — is absent from Codex.

The work is dominated by (a) deepenings of the five templated 06.01.x stubs (highest priority — they currently fail equivalence at the prose-quality layer entirely) and (b) new units for the distinctive Ahlfors back-half block (Chapters 5-8). New-unit volume is larger than peer audits: ~12-15 NEW vs Forster's 7 and Donaldson's 4.

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§1 What Ahlfors's book is for

The canonical graduate-level introduction to one-complex-variable analysis. Where Stein-Shakarchi anchors the real-analysis-flavoured modern style, Conway the encyclopaedic measure-theoretic style, and Rudin's Real and Complex the terse-axiomatic style, Ahlfors anchors the geometric / function-theoretic / hyperbolic-metric tradition. It is the book PhD students cite when they say "complex analysis" without qualification — the originator-text for the modern presentation of normal families, Schwarz-Pick on the disc, and Picard's theorems.

8 chapters, ~331 pages:

• Ch. 1 Complex Numbers (~25 pp). Algebra, polar form, roots of unity; Riemann sphere ${\mathbb{C}}_{\infty }$ via stereographic projection; spherical chordal metric.
• Ch. 2 Complex Functions (~50 pp). Topology, analyticity, Cauchy-Riemann; conformal maps; elementary functions ($e^z$, $\log z$, $z^{\alpha }$, trig); branches; Möbius transformations, cross-ratio, symmetry; upper half-plane ↔ disc.
• Ch. 3 Analytic Functions as Mappings (~60 pp). Elementary conformal mapping; harmonic functions (Laplacian, mean-value, Poisson integral, reflection); complex integration intro; index of a closed curve; Cauchy's integral formula.
• Ch. 4 Complex Integration (~70 pp; densest chapter). General Cauchy's theorem (homological form); residue calculus + real- integral evaluation; argument principle + Rouché; maximum modulus + Schwarz lemma + Schwarz-Pick; Phragmén-Lindelöf; harmonic functions, Dirichlet on disc.
• Ch. 5 Series and Product Developments (~65 pp). Power / Taylor / Laurent series; Mittag-Leffler partial fractions; infinite products; Weierstrass factorization theorem; gamma function $\Gamma (z)$; Riemann zeta $\zeta (s)$, analytic continuation, functional equation; normal families (Montel).
• Ch. 6 Conformal Mapping, Dirichlet's Problem (~35 pp). Riemann mapping theorem (proof via normal families); boundary behaviour; Schwarz-Christoffel formula; Dirichlet via Perron's method; harmonic measure.
• Ch. 7 Elliptic Functions (~30 pp). Doubly-periodic functions; Weierstrass $\wp$, $({\wp }^{\prime }{)}^2=4{\wp }^3-g_2\wp -g_3$; theta functions; modular function $\lambda$, modular group, $j$- invariant.
• Ch. 8 Global Analytic Functions (~30 pp). Analytic continuation along curves; monodromy theorem; Riemann surface of an analytic function; algebraic functions; Picard's little + great theorems; Schottky, Bloch; uniformization closing arc.

Distinctive Ahlfors choices (the load-bearing list, Codex-relevance flagged):

1. Geometric throughout. Conformality drives Ch. 2-3; the Schwarz-Pick hyperbolic metric on the disc is woven into Schwarz lemma so Picard, Schottky, Bloch, Riemann mapping all become hyperbolic-isometry / contraction arguments. Codex: zero hyperbolic-metric content.
2. Riemann sphere from Chapter 1. ${\mathbb{C}}_{\infty }=S^2$ in §1.4; chordal metric immediately. Möbius transformations are chordal isometries (up to rotation). Codex: no Riemann-sphere unit.
3. Möbius transformations as a category. $\mathrm{PSL}2(\mathbb{C}) \curvearrowright \mathbb{C}\infty$, cross-ratio, classification (parabolic / elliptic / loxodromic / hyperbolic). Bridge to hyperbolic geometry. Codex: absent.
4. Cauchy's theorem in homological form. Cycles, winding numbers, Dixon proof. Codex 06.01.02 is templated — no proof, no Dixon.
5. Argument principle + Rouché in one section as residue corollaries on $f^{\prime }/f$. Codex: absent.
6. Schwarz lemma → Schwarz-Pick → hyperbolic geometry. Foundational for Picard. Codex: absent.
7. Phragmén-Lindelöf for unbounded domains. Codex: absent.
8. Harmonic functions + Dirichlet (Poisson / Perron). Half of why complex analysis matters in PDE / fluid / EM applications. Codex: zero harmonic content.
9. Weierstrass factorization on $\mathbb{C}$. Codex's 06.09.05-cousin-ii mentions the RS-corollary; the plane theorem absent.
10. Gamma function $\Gamma (z)$ — full development. Single most-cited example of analytic continuation. Codex: absent.
11. Riemann zeta $\zeta (s)$, analytic continuation, functional equation. Codex: absent.
12. Normal families / Montel. Used for Riemann mapping (Ch. 6) and Picard (Ch. 8). Codex: absent.
13. Schwarz-Christoffel for polygons. PDE / fluid / EM tool. Codex: absent.
14. Picard little + great. Capstone of one-complex-variable. Codex: absent.
15. Schottky / Bloch. Quantitative Picard. Codex: absent.
16. Weierstrass $\wp$ + modular function $\lambda /j$. Codex's 06.06.05-theta-function is templated; no $\wp$ unit, no modular-form unit.
17. Monodromy theorem (Ch. 8 §1). Codex 06.01.04 is templated and does not state monodromy.
18. ~280 exercises, heavy load-bearing volume. Codex has the templated 7-block on each shipped 06.01.x — none solve a real Ahlfors exercise.

Ahlfors does not cover (scope clarity): $H^p$ Hardy spaces beyond Phragmén-Lindelöf, several-complex-variables Hartogs (FT 06.07 territory), automorphic forms beyond modular function (Farkas-Kra territory), Riemann-Roch on compact surfaces (FT 1.07 — Donaldson / Forster / Miranda).

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§2 Coverage table (Codex vs Ahlfors)

Existing Codex units relevant to Ahlfors, from find content -name "*.md" | sort:

• 06.01-complex-analysis/ (6 units, 5 templated): 06.01.01-holomorphic-function (only one with genuine prose), 06.01.02-cauchy-integral-formula (templated), 06.01.03-residue-theorem (templated), 06.01.04-analytic-continuation (templated), 06.01.05-meromorphic-function (templated), 06.01.06-riemann-mapping-theorem (templated).
• 06.02-coverings/: 06.02.02-branched-coverings, 06.02.03-riemann-s-existence-theorem-for-algebraic-curves.
• 06.03-riemann-surfaces/: 06.03.01-riemann-surface, 06.03.03-uniformization-theorem.
• 06.06-jacobians/: 06.06.01-holomorphic-one-form, 06.06.05-theta-function (templated).
• 06.07-several-variables/: 06.07.01-holomorphic-several-variables, 06.07.02-hartogs-phenomenon.
• 06.09-stein/: 06.09.05-cousin-ii-multiplicative (touches Weierstrass factorisation as Cousin-II corollary).
• 02.11-functional-analysis/ (background only): 02.11.05-compact-operators, 02.11.08-hilbert-space.

✓ = covered at Ahlfors-equivalent depth, △ = topic present but Codex unit shallower / templated, ✗ = not covered.

Chapter 1 — Complex Numbers

Ahlfors topic	Codex unit	Status	Note
§1.1-1.2 Complex algebra, polar form, roots of unity	—	✗	Gap. Belongs in 00-precalc or new 06.01.00.
§1.3 The complex plane as a metric space	partial in 02.01.05-metric-space	△	Generic metric-space coverage; complex specifics absent.
§1.4 Riemann sphere ${\mathbb{C}}_{\infty }$ via stereographic projection	—	✗	Gap (P1). Foundational; cited by every Möbius / global discussion.
§1.4 Spherical chordal metric	—	✗	Gap. Same unit.

Chapter 2 — Complex Functions

Ahlfors topic	Codex unit	Status	Note
§2.1 Topology of $\mathbb{C}$, limits, sequences	partial in 02.01.05	△	Generic.
§2.1 Analytic functions, Cauchy-Riemann	06.01.01-holomorphic-function	△	Definition shipped at depth; CR equations not unpacked at Ahlfors's level (real-and-imaginary-part decomposition; harmonic conjugate).
§2.2 Conformal mappings	mention in 06.01.01	△	Mention only; no dedicated unit.
§2.3 Elementary functions ($e^z$, $\log z$, $\sin z$, $\cos z$, branches)	—	✗	Gap (P1). Foundational; sets up branches discussion needed for monodromy / analytic continuation.
§2.3 Branches of $\log$ and $z^{\alpha }$	—	✗	Gap (P1). Forward-declared by 06.01.04-analytic-continuation but never built.
§2.4 Möbius (linear-fractional) transformations; cross-ratio	—	✗	Gap (P1). Hyperbolic geometry foundation.
§2.4 Symmetry / inversion on the sphere; classification (parabolic / elliptic / loxodromic)	—	✗	Gap (P1). Same unit.
§2.4 Upper half-plane ↔ disc model	—	✗	Gap. Half of Schwarz-Pick.

Chapter 3 — Analytic Functions as Mappings

Ahlfors topic	Codex unit	Status	Note
§3.1 Elementary conformal mapping (Möbius + elementary functions in detail)	—	✗	Cross-references Ch 2 + new Möbius unit; depth gap.
§3.2 Harmonic functions; Laplacian; mean-value property	—	✗	Gap (P1 — Ahlfors-distinctive). Codex has zero harmonic-function content.
§3.2 Poisson integral on the disc	—	✗	Gap (P1). Same unit or sibling.
§3.2 Schwarz reflection principle	—	✗	Gap (P2).
§3.3 Line integrals, primitives, Cauchy's theorem (rectangle / disc)	partial in 06.01.02	△	Templated; no proof at Ahlfors level (Goursat triangle; primitive existence on simply-connected domains).
§3.4 Index of a closed curve / winding number	—	✗	Gap (P1). Cited by 06.01.03-residue-theorem but not built.
§3.4 Cauchy's integral formula	06.01.02-cauchy-integral-formula (templated)	△	Templated stub — DEEPEN required.

Chapter 4 — Complex Integration

Ahlfors topic	Codex unit	Status	Note
§4.1 General Cauchy's theorem (homological version, Dixon proof)	partial in 06.01.02	△	Templated; DEEPEN essential. Ahlfors's homological framing absent.
§4.1 Cycles, winding numbers, homology in domains	—	✗	Gap (P1).
§4.2 Cauchy integral formula for cycles	—	✗	Gap. Same unit as homological Cauchy.
§4.3 Residue calculus + evaluation of real integrals	06.01.03-residue-theorem (templated)	△	Templated stub — DEEPEN essential.
§4.3 Argument principle; Rouché's theorem	—	✗	Gap (P1).
§4.4 Maximum modulus principle; Schwarz lemma	partial / mention	✗	Gap (P1). Mentioned in 06.01.01 Master only.
§4.4 Schwarz-Pick / hyperbolic metric on the disc	—	✗	Gap (P1 — Ahlfors-distinctive).
§4.5 Phragmén-Lindelöf principle	—	✗	Gap (P2).
§4.6 Harmonic functions (Laplacian, mean value, Poisson)	—	✗	Gap (P1).
§4.7 Schwarz reflection principle	—	✗	Gap (P2). Often grouped with §3.2.

Chapter 5 — Series and Product Developments

Ahlfors topic	Codex unit	Status	Note
§5.1 Power series, radius of convergence	—	✗	Gap (P2). Foundational; might fit under 00-precalc/02.07-series or new unit.
§5.1 Taylor and Laurent series	partial in 06.01.05 Master	△	Templated mention; expansion theorem not stated at depth.
§5.2 Partial-fraction (Mittag-Leffler) on $\mathbb{C}$	partial via 06.09.04-cousin-i (RS case)	△	Plane case (Ahlfors's level) absent; only RS generalisation shipped.
§5.2 Infinite products	—	✗	Gap (P1).
§5.2 Weierstrass factorization theorem (entire functions)	partial via 06.09.05-cousin-ii	△	Cousin-II generalisation shipped; classical plane statement absent.
§5.2 Gamma function $\Gamma (z)$ — full development	—	✗	Gap (P1 — Ahlfors-distinctive).
§5.4 Riemann zeta function $\zeta (s)$, analytic continuation, functional equation	—	✗	Gap (P1 — Ahlfors-distinctive).
§5.5 Normal families; Montel's theorem	—	✗	Gap (P1). Used in Ch 6 + Ch 8 Picard.

Chapter 6 — Conformal Mapping, Dirichlet's Problem

Ahlfors topic	Codex unit	Status	Note
§6.1 Riemann mapping theorem (proof via normal families)	06.01.06-riemann-mapping-theorem (templated)	△	Templated stub — DEEPEN essential. Currently no proof, no statement at Ahlfors depth.
§6.1 Boundary behaviour (Carathéodory)	—	✗	Gap (P3).
§6.2 Schwarz-Christoffel formula	—	✗	Gap (P1).
§6.4 Dirichlet problem on general domain (Perron's method)	—	✗	Gap (P2). Depends on harmonic-function unit.
§6.5 Harmonic measure	—	✗	Gap (P3).

Chapter 7 — Elliptic Functions

Ahlfors topic	Codex unit	Status	Note
§7.1 Doubly-periodic functions; period lattice	partial via 06.06.02-period-matrix	△	RS-generalised; classical $\mathbb{C}/\Lambda$ unit absent.
§7.2 Weierstrass $\wp$ function; differential equation $({\wp }^{\prime }{)}^2=4{\wp }^3-g_2\wp -g_3$	—	✗	Gap (P1).
§7.3 Theta functions ${\vartheta }_i$ on $\mathbb{H}$	partial via 06.06.05-theta-function (templated)	△	Templated; no ${\vartheta }_i$ split, no Jacobi product.
§7.4 Modular function $\lambda$, modular group, $j$-invariant	—	✗	Gap (P2).
§7.4 Fundamental domain of ${\mathrm{PSL}}_2(\mathbb{Z})$	—	✗	Gap (P2). Same unit.

Chapter 8 — Global Analytic Functions

Ahlfors topic	Codex unit	Status	Note
§8.1 Analytic continuation along curves	partial in 06.01.04 (templated)	△	Templated; DEEPEN essential. Curve-continuation explicit; monodromy not stated.
§8.1 Monodromy theorem	—	✗	Gap (P1). Belongs in 06.01.04 deepening.
§8.2 Riemann surface of an analytic function (sheet construction)	partial via 06.03.01	△	RS as object shipped; sheet / fibred construction from analytic function absent.
§8.2 Algebraic functions $P(z,w)=0$	06.02.03-riemann-s-existence-theorem-for-algebraic-curves	✓	Forster-batch shipped.
§8.3 Picard's little theorem (entire ⇒ omits ≤ 1 value)	—	✗	Gap (P1 — Ahlfors-distinctive).
§8.3 Picard's great theorem (essential singularity)	—	✗	Gap (P1).
§8.3 Schottky's theorem	—	✗	Gap (P3).
§8.3 Bloch's theorem	—	✗	Gap (P3).

Aggregate coverage estimate (vs Ahlfors)

• Theorem layer: ~30% topic, ~12% Ahlfors-proof-depth (most shipped units are templated). After P1 deepenings + P1 new units: topic ~85%, proof-depth ~70%. After P1+P2: ~93% / ~85%.
• Exercise layer: ~3% (Ahlfors's ~280 exercises vs templated 7-block on 5 of 6 shipped units, none of which solve real Ahlfors exercises). Dedicated exercise pack 06.01.E1 / 06.01.E2 required at P3.
• Worked-example layer: ~10%. The contour-integration worked examples that anchor Chapter 4 are absent; a residue-pack / contour-pack unit fills this at P2.
• Notation layer: ~50% aligned. Ahlfors uses $\Omega$ (region), ${\mathbb{C}}_{\infty }$ (Riemann sphere), $\wp (z;{\omega }_1,{\omega }_2)$, $\Gamma (z)/B(p,q)$, $\zeta (s)$, $\lambda (\tau )/j(\tau )$. Codex notation drift mild; notation/ahlfors.md low-priority.
• Sequencing layer: ~30%. Ahlfors's chapter chain (Ch 1 sphere → Ch 2 elementary functions / Möbius → Ch 3 conformal + harmonic → Ch 4 integration / Schwarz → Ch 5 normal families / $\Gamma$ / $\zeta$ → Ch 6 Riemann mapping / Schwarz-Christoffel / Dirichlet → Ch 7 $\wp$ / modular → Ch 8 Picard) currently has ~3 of 8 nodes shipped. Most prerequisites missing.
• Intuition layer: ~25%. Ahlfors's geometric-Schwarz-Pick framing absent; templated stubs offer no intuition.
• Application layer: ~15%. The standard contour-integration tour (∫ from $-\infty$ to $\infty$ of rational, trig, $e^{ix}$ × rational) is what makes complex analysis applied; Codex has zero coverage.

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§3 Gap punch-list (P3-lite — units to write or deepen, priority-ordered)

The shape of this audit differs from peer surfaces audits: Ahlfors is the prerequisite, not the surface. Codex's 06.01-complex-analysis/ chapter currently fails equivalence on two axes — (a) five of six shipped units are templated v0.5 Strand C/D stubs without genuine prose (DEEPEN required), and (b) Ahlfors's back-half block (Ch 5-8) is entirely absent (NEW required). Mid-volume new-unit count (~12-15 NEW units) and heavy DEEPEN load (~5 templated stubs replaced

• 3 cross-chapter Master sections enriched).

Priority 1 — Templated-stub deepenings + Ahlfors-distinctive headlines

The five templated 06.01.x stubs currently fail equivalence at the prose-quality layer entirely. These rewrites are required before any new unit can claim Ahlfors as a tier_anchor honestly. The "Ahlfors- distinctive" new units are the back-half headlines.

1. 06.01.02 Cauchy integral formula. [DEEPEN] Replace templated stub with Ahlfors §4.1-4.2: rectangle / disc Cauchy (Goursat triangle proof); homological Cauchy on cycles via Dixon; Cauchy integral formula on cycles; Cauchy estimates → Liouville → FTA. ~2000 words. Originator: Cauchy 1825 Mémoire sur les intégrales définies prises entre des limites imaginaires; Goursat 1900; Dixon 1971 (Proc. AMS 29).

2. 06.01.03 Residue theorem. [DEEPEN] Ahlfors §4.3: residue at pole / infinity; residue theorem on cycles; canonical contour- integration tour (∫ rational over $\mathbb{R}$; rational × trig; rational × $e^{i\alpha x}$; indented contours; keyhole contours); argument principle $\oint f^{\prime }/f=N-P$; Rouché. ~2400 words; largest deepening. Originator: Cauchy 1826; Rouché 1862.

3. 06.01.04 Analytic continuation. [DEEPEN] Ahlfors §8.1: continuation along a curve; between overlapping discs; monodromy theorem on simply-connected domain; examples $\sqrt{z}$, $\log z$. ~1700 words. Originator: Weierstrass c. 1842 lectures; Riemann 1857.

4. 06.01.05 Meromorphic function. [DEEPEN] Ahlfors §5.1-5.2: poles / zeros; isolated-singularity classification (removable / pole / essential); Casorati-Weierstrass; Laurent expansion with proof and uniqueness; meromorphic = local ratio of holomorphic. ~1700 words. Originator: Weierstrass; Casorati 1868.

5. 06.01.06 Riemann mapping theorem. [DEEPEN] Ahlfors §6.1: statement; proof via normal families (extremal $\sup |f^{\prime }(z_0)|$ solved by Montel + Hurwitz); uniqueness up to disc Möbius; explicit maps (half-plane, strip, slit-disc). ~2000 words. Depends on item 13 (Montel). Originator: Riemann 1851 statement; Koebe 1907, Carathéodory 1912 proof.

6. 06.01.07 Riemann sphere ${\mathbb{C}}_{\infty }$. [NEW] Ahlfors §1.4. Stereographic projection $S^2\setminus \{N\}\cong \mathbb{C}$; ${\mathbb{C}}_{\infty }$ as compact genus-0 Riemann surface; chordal metric $d(z,w)=2|z-w|/\sqrt{(1+|z{|}^2)(1+|w{|}^2)}$; meromorphic functions ↔ holomorphic maps to ${\mathbb{C}}_{\infty }$. ~1500 words. Originator: Riemann 1851. Foundational for every Möbius / global discussion.

7. 06.01.08 Möbius (linear-fractional) transformations. [NEW] Ahlfors §2.4. $\mathrm{PSL}2(\mathbb{C}) \curvearrowright \mathbb{C}\infty$; classification by trace (parabolic / elliptic / hyperbolic / loxodromic); cross-ratio as Möbius invariant; circle-line preservation; symmetry / inversion. ~1800 words. Worked: $\mathbb{H}\to \mathbb{D}$ map; three-points-to-$\{0,1,\infty \}$. Originator: Möbius 1855; Klein 1872. Foundational for Schwarz-Pick, Schwarz-Christoffel, modular group.

8. 06.01.09 Elementary complex functions: $e^z,\log z,z^{\alpha }$, trig. [NEW] Ahlfors §2.3. Power-series $e^z$; periodicity $2\pi i$; branches of $\log z$ on simply-connected $\Omega \subset \mathbb{C}^*$;principalbranch;$z^\alpha$;$\sin z, \cos z$; hyperbolic functions; inverse trig as branches. ~1700 words.

9. 06.01.10 Cauchy-Riemann equations + harmonic conjugate. [NEW] Ahlfors §2.1, §3.2. CR system; equivalent to $\partial f/\partial \bar z = 0$;$\Delta u = \Delta v = 0$; harmonic-conjugate existence on simply-connected $\Omega$; Looman-Menchoff converse sketch. ~1500 words.

10. 06.01.11 Harmonic functions on the plane. [NEW] Ahlfors §3.2, §4.6. Laplacian; mean-value property; Poisson integral on the disc; harmonic maximum principle; harmonic Liouville. ~1800 words. Originator: Laplace 1782; Poisson 1820. Foundation for Schwarz reflection, Picard, Dirichlet.

11. 06.01.12 Maximum modulus + Schwarz lemma. [NEW] Ahlfors §4.4. Maximum modulus on bounded $\Omega$; Schwarz lemma ($f:\mathbb{D}\to \mathbb{D},f(0)=0$ ⇒ $|f|\le |z|$; equality ⇒ rotation); Schwarz-Pick ($f: \mathbb{D} \to \mathbb{D}$ contracts hyperbolic metric). ~1800 words. Originator: Schwarz 1869; Pick 1916. Foundation for Picard.

12. 06.01.13 Argument principle and Rouché. [NEW] Ahlfors §4.3. $\frac{1}{2\pi i}\oint f^{\prime }/f=N-P$; Rouché: $|f-g|<|g|$ on $\partial \Omega$ ⇒ equal zero counts. Worked: zero count of $z^7+4z^3+11$ in $|z|<1$; Rouché-FTA. ~1300 words.

13. 06.01.14 Normal families and Montel's theorem. [NEW] Ahlfors §5.5. Locally uniformly bounded family ⇒ normal; Vitali; Hurwitz (limit of nonvanishing holomorphic is nonvanishing or $\equiv 0$). ~1700 words. Cited by item 5 deepening + items 19,

    20. Originator: Montel 1907 (Ann. ENS); Vitali 1903.

14. 06.01.15 Gamma function $\Gamma (z)$. [NEW] Ahlfors §5.2. Euler integral on $\Re z>0$; functional equation $\Gamma(z+1) = z\Gamma(z)$continuesto$\mathbb{C} \setminus {0, -1, -2, \ldots}$; Weierstrass product $1/\Gamma(z) = z e^{\gamma z} \prod (1 + z/n) e^{-z/n}$;\ast \ast reflection\ast \ast$\Gamma(z)\Gamma(1-z) = \pi/\sin\pi z$; Stirling. ~2000 words. Originator: Euler 1729 (Goldbach letter); Gauss 1812; Weierstrass 1856. Single most-cited example of analytic continuation.

15. 06.01.16 Riemann zeta function $\zeta (s)$. [NEW] Ahlfors §5.4. Dirichlet series on $\Re s>1$; Euler product over primes; analytic continuation to $\mathbb{C}\setminus \{1\}$ via integral representation $\zeta(s)\Gamma(s) = \int_0^\infty t^{s-1}/(e^t - 1) dt$;\ast \ast functionalequation\ast \ast$\xi(s) = \xi(1-s)$; pointer to RH. ~2200 words. Originator: Euler 1737; Riemann 1859 Über die Anzahl der Primzahlen unter einer gegebenen Grösse. Originator-prose mandatory.

16. 06.01.17 Weierstrass factorization theorem. [NEW] Ahlfors §5.2. Entire $f$ with zeros $\{a_n\}$ factors $f = z^m e^{g(z)} \prod E_{p_n}(z/a_n)$ via Weierstrass primary factors; convergence via $\sum 1/|a_n{|}^{p+1}<\infty$. ~1700 words. Originator: Weierstrass 1876. (Plane prequel to 06.09.05 Cousin II.)

17. 06.01.18 Mittag-Leffler theorem on $\mathbb{C}$. [NEW] Ahlfors §5.2. Prescribed principal parts $P_n$ at $a_n\to \infty$ ⇒ meromorphic realisation; convergence via polynomial subtraction. ~1300 words. Originator: Mittag-Leffler 1884 (Acta Math. 4). (Plane prequel to 06.09.06-RS-case.)

18. 06.01.19 Schwarz-Christoffel formula. [NEW] Ahlfors §6.2. $\mathbb{H}\to$ polygon: $f^{\prime }(z)=K{\prod }_k(z-x_k{)}^{{\alpha }_k-1}$; rectangle, triangle, regular-polygon worked examples. ~1700 words. Originator: Schwarz 1869; Christoffel 1867. PDE / fluid-mechanics tool.

19. 06.01.20 Picard's little theorem. [NEW] Ahlfors §8.3. Non- constant entire ⇒ omits at most one value. Proof via Montel + Schwarz-Pick on the modular $\lambda$. ~1600 words. Originator: Picard 1879 (CRAS 88). Depends on items 11, 13, 26.

20. 06.01.21 Picard's great theorem. [NEW] Ahlfors §8.3. Essential singularity ⇒ image takes every value (≤ 1 exception) infinitely often. Proof via Montel of $\{f((z-z_0)/r)\}$ + little Picard. ~1500 words. Originator: Picard 1880. Depends on items 13, 19. Capstone of one-complex-variable.

Priority 2 — Ahlfors-distinctive depth (Ch 4-7)

Topics that are not strictly equivalence-blockers but close prominent Ahlfors gaps and deliver the bulk of the chapter's promised pedagogy.

21. 06.01.22 Phragmén-Lindelöf principle. [NEW] Ahlfors §4.5. Extension of maximum modulus to unbounded domains via auxiliary bounded subharmonic comparison. Worked: half-plane / sector / strip versions. ~1500 words. Originator: Phragmén-Lindelöf 1908 (Acta Math. 31).

22. 06.01.23 Schwarz reflection principle. [NEW] Ahlfors §3.2, §4.7. Holomorphic on upper half-plane, real-valued on $\mathbb{R}$ boundary segment ⇒ extends across to lower half-plane via $f(\bar{z})=\overline{f(z)}$. Generalised reflection across arcs / circles. ~1300 words. Originator: Schwarz 1870.

23. 06.01.24 Dirichlet problem on the disc + Perron's method. [NEW] Ahlfors §3.2, §6.4. Existence of harmonic function with prescribed boundary values: disc case via Poisson integral (item 11); general domain via Perron's method (sup over subharmonic subsolutions). Boundary regularity (barriers). ~2000 words. Originator: Dirichlet (1840s lectures); Perron 1923 (Math. Z. 18). Half of why complex analysis matters in PDE.

24. 06.01.25 Weierstrass $\wp$ function. [NEW] Ahlfors §7.2. Period lattice $\Lambda =\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2$; $\wp (z)=1/z^2+{\sum }_{\omega \ne 0}(1/(z-\omega {)}^2-1/{\omega }^2)$; differential equation $({\wp }^{\prime }{)}^2=4{\wp }^3-g_2\wp -g_3$; $\mathbb{C}/\Lambda$ as elliptic curve $y^2=4x^3-g_{2}x-g_3$; field of elliptic functions $=\mathbb{C}(\wp ,{\wp }^{\prime })$. ~2000 words. Originator: Weierstrass 1862-63 lectures (published Eisenstein 1840s earlier with $\wp$-like series). Classical-plane analogue of 06.06.05-theta-function.

25. 06.01.26 Modular function $\lambda$ and $j$-invariant. [NEW] Ahlfors §7.4. Modular group ${\mathrm{PSL}}_2(\mathbb{Z})$ on $\mathbb{H}$; fundamental domain; $\lambda (\tau )$ as modular function for $\Gamma (2)$ (omits 0, 1, $\infty$); $j$-invariant $j(\tau )$ for ${\mathrm{PSL}}_2(\mathbb{Z})$; isomorphism $\mathbb{H}/\Gamma (2)\cong \mathbb{C}\setminus \{0,1\}$ used in classical Picard proof. ~1700 words. Originator: Klein 1879 (j-invariant); Picard 1879 (use in Picard's theorem).

26. 06.01.27 Power series and Laurent series (proofs). [NEW] Ahlfors §5.1. Radius of convergence $1/\mathrm{limsup}|a_n{|}^{1/n}$; holomorphic ⇔ locally power-series-expandable; Laurent expansion theorem on annulus with proof; uniqueness. ~1500 words. (Could be folded into 06.01.05 deepening if scope pressure; standalone preferred for citation depth.)

27. 06.01.28 Index / winding number of a closed curve. [NEW] Ahlfors §3.4. $n(\gamma, z_0) = \frac{1}{2\pi i} \oint_\gamma dw/(w - z_0)$; integer-valued; locally constant; computation examples; relation to fundamental group of $\mathbb{C} \setminus {z_0}$. ~1100 words. Foundation for homological Cauchy. Could be folded into 06.01.02 deepening; standalone preferred.

Priority 3 — Composite / cross-chapter deepenings + exercise pack

28. Deepen 06.01.01-holomorphic-function. [ENRICH] Already at depth; add Ahlfors §1-2 explicit references for Cauchy-Riemann equations, harmonic-conjugate corollary, branches discussion forward-pointer to item 8.

29. 06.01.E1 Contour integration exercise pack. [NEW] ~30 Ahlfors-style contour integrals: rational over $\mathbb{R}$, rational × trig, rational × $e^{i\alpha x}$, indented contours, keyhole contours, mixed-branch integrals. ~2500 words. Closes the worked-example layer for Ch 4.

30. 06.01.E2 Ahlfors exercise pack (selected). [NEW] ~25 Ahlfors exercises (Ch 1-8 spread) with full solutions. ~3000 words. Closes the exercise-layer to Ahlfors.

31. Deepen 06.06.05-theta-function. [ENRICH/DEEPEN composite] Already on Forster/Donaldson plan as composite item; add Ahlfors §7.3 plane-case framing (Jacobi ${\vartheta }_i$, the four classical theta functions, product representations).

32. Deepen 06.03.03-uniformization-theorem. [ENRICH] Add Ahlfors §8 closing-arc reference; Ahlfors's framing of uniformization as the Picard / monodromy capstone.

33. notation/ahlfors.md notation crosswalk. [NEW — optional] $\Omega$ (region), ${\mathbb{C}}_{\infty }$ (Riemann sphere), $\wp (z;{\omega }_1,{\omega }_2)$, $\Gamma$, $\zeta$, $\lambda$, $j$. ~400 words. Defer unless 06.06-side $X/M/\Sigma$ cleanup undertaken.

Priority 4 — Pointers / surveys, optional

34. 06.01.29 Schottky's and Bloch's theorems. [NEW — optional] Ahlfors §8.3. Quantitative Picard refinements. ~1300 words. Cite if items 19, 20 don't already absorb. P4 unless gauge-theory / value-distribution downstream demands.

35. 06.01.30 Riemann-Hurwitz for plane meromorphic / sphere maps. [NEW — optional pointer] Cross-link to 06.05.03-riemann-hurwitz-formula via the sphere-as-genus-0 case. P4.

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§4 Implementation sketch (P3 → P4)

Minimum Ahlfors-equivalence batch = priority 1 (items 1-20): 5 deepenings + 15 new units, all Ahlfors-only. Production estimate:

• ~1.5 h per deepening of templated stub (research + replace prose + validate at 27/27).
• ~3 h per typical new unit; ~4 h for items 14, 15 (gamma, zeta — load-bearing originator-prose units), item 5 deepening (Riemann mapping with full Montel proof), item 2 deepening (residue calculus
  • argument principle + Rouché).
• Priority 1 totals: 5 deepenings × 1.5 h + 11 typical new × 3 h + 4 large × 4 h = ~56 h.
• Priority 1+2 totals: 56 h + 7 P2 units × 3 h = ~77 h.
• Priority 1+2+3 totals: 77 h + 3 P3 packs × 4 h + 2 enrichments × 1 h = ~91 h.

At 4-6 parallel production agents, priority-1 fits in a 3-4 day window. This is the largest single-chapter audit yet — comparable to Hartshorne's depth gap, not Cannas's.

Batch structure.

• Batch A (template-replace, items 1, 2, 3, 4, 5, ~10 h): the five DEEPEN items. Highest priority: cleans the existing chapter surface so subsequent units can cite shipped prose without forward-templated-stub references. Produce together; load-bearing for everything downstream.
• Batch B (Ahlfors front-half foundations, items 6, 7, 8, 9, 10, ~13 h): Riemann sphere → Möbius → elementary functions → Cauchy-Riemann → harmonic functions. Depends on Batch A for 06.01.01 enrichment references. Foundational for Batch C-D.
• Batch C (Ahlfors integration headlines, items 11, 12, 13, 17, ~9 h): maximum modulus / Schwarz / Schwarz-Pick / argument principle / Rouché / Mittag-Leffler. Depends on Batch B.
• Batch D (back-half headlines: gamma, zeta, Weierstrass factorization, Montel, Riemann mapping retake, items 14, 15, 16, 18, ~14 h): Ahlfors-distinctive Ch 5 block + Schwarz-Christoffel. Item 5 deepening cites item 18 (Montel) — produce in same batch with forward-citation discipline.
• Batch E (Picard capstone, items 19, 20, ~5 h): depends on Batches B-D. Closing arc.
• Batch F (P2 polish + exercise packs, ~25 h): items 21-30. Optional / phased after Batch A-E.

Originator-prose targets (priority-1 Master sections cite originator

• Ahlfors). Most originators listed inline above; the load-bearing originator-voice-mandatory units are: item 6 (Riemann 1851); item 14 (Euler 1729 Goldbach letter / Gauss 1812 / Weierstrass 1856); item 15 (Euler 1737 / Riemann 1859 Über die Anzahl der Primzahlen unter einer gegebenen Grösse); item 19 (Picard 1879, CRAS 88, 1024-1027). Other items cite standard primary sources (Cauchy 1825/1826, Schwarz 1869, Pick 1916, Möbius 1855, Klein 1872, Picard 1880, Montel 1907, Mittag- Leffler 1884, Weierstrass 1876, Christoffel 1867, Goursat 1900, Dixon 1971, Casorati 1868, Koebe 1907, Carathéodory 1912, Laplace 1782, Poisson 1820, Vitali 1903).

Notation crosswalk. Mostly aligned with shipped 06.01.x. Defer notation/ahlfors.md (item 33) unless chapter-wide pass undertaken.

DAG edges to add (priority-1 + 2):

• 06.01.07 ← 06.01.01
• 06.01.08 ← {06.01.07, 06.01.01}
• 06.01.09, 06.01.10 ← 06.01.01
• 06.01.11 ← {06.01.10, 06.01.02}
• 06.01.12 ← {06.01.02, 06.01.11}
• 06.01.13 ← 06.01.03
• 06.01.14 ← {06.01.01, 06.01.02, 02.01.05}; → 06.01.06-deepened, items 19, 20
• 06.01.15, 06.01.16, 06.01.17 ← 06.01.05; 06.01.16 → 06.09.05; 06.01.17 → 06.09.04
• 06.01.18 ← {06.01.06, 06.01.08}
• 06.01.19 ← {06.01.12, 06.01.14, 06.01.26}
• 06.01.20 ← {06.01.14, 06.01.19}
• 06.01.22 ← 06.01.12; 06.01.23 ← {06.01.11, 06.01.10}; 06.01.24 ← 06.01.11
• 06.01.25 ← {06.01.05, 06.01.16}; lateral to 06.06.05
• 06.01.26 ← {06.01.08, 06.01.25}

Composite Ahlfors + Donaldson + Forster batch. Item 16 (Weierstrass factorization on $\mathbb{C}$) and item 17 (Mittag-Leffler on $\mathbb{C}$) are the plane prequels to 06.09.05-cousin-ii and 06.09.06-Mittag-Leffler-on-RS (already shipped from the Forster batch). Producing items 16, 17 retroactively strengthens the Forster chain. Item 5 deepening (Riemann mapping with Montel proof) is the plane prequel to the Donaldson uniformization deepening; co-cite.

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§5 What this plan does NOT cover

• Line-number-level inventory of theorems / exercises across all 8 chapters. Defer unless priority-1+2 expands.
• Ahlfors's ~280 exercises vs. templated 7-block. Items 29-30 close ~50; remainder deferred to a v0.7+ comprehensive complex-analysis exercise pack.
• Donaldson-distinctive RS material (Hodge on RS, RS-PDE, Gauss-Manin / VHS, Schottky on RS). Donaldson plan owns.
• Forster-distinctive sheaf-theoretic RS material (Stein 1-d, Cousin I/II on RS, Behnke-Stein, Schwartz finiteness). Forster owns; plane prequels (items 16, 17) owned here.
• Several-complex-variables Hartogs / Stein (FT 06.07 territory; shipped).
• Number-theoretic $\zeta$ beyond the functional equation (RH, PNT, $L$-functions). Item 15 stops at Ahlfors Ch 5; Manin / Apostol-NT territory deferred.
• Modular-form machinery beyond $\lambda ,j$ (higher weight, Eisenstein, Hecke). Diamond-Shurman / Serre Cours d'arithmétique territory.
• $H^p$ Hardy spaces, BMO, Bergman, Beurling. Garnett / Rudin RAC territory.
• Nevanlinna value-distribution theory. Quantitative defect / characteristic functions deferred.
• Quasiconformal maps, Beltrami, Teichmüller. Ahlfors's other book (Lectures on Quasiconformal Mappings); out of scope.
• Worked-example densification beyond items 29-30; v0.7+ pass.
• Notation crosswalk (item 33) deferred unless chapter-wide pass.

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§6 Acceptance criteria for FT equivalence (Ahlfors)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4 and §9, the book is at equivalence-coverage when:

• ≥95% of Ahlfors's named theorems map to Codex units at Ahlfors-equivalent proof depth (currently ~12% — five of six shipped units are templated; after Batch A 5 deepenings, ~35%; after priority-1 (Batches A-E), ~85%; after priority-1+2, ~93%; after priority-1+2+3, ~96%).
• ≥80% of Ahlfors's exercises have a Codex equivalent (currently ~3% — the templated 7-block on five of six units; closing this requires items 29-30 plus the future v0.7 comprehensive complex-analysis exercise pack).
• ≥90% of Ahlfors's worked examples reproduced in some Codex unit (currently ~10% — the contour-integration tour and the sphere / Möbius worked maps absent; priority-1 deepenings + items 29-30 bring this to ~90%).
• Notation alignment recorded inline in priority-1 deepenings + new Master sections; optional notation/ahlfors.md (item 33) deferred.
• DAG prerequisites arrows for every Ahlfors chapter dependency (Ch 1 sphere → Ch 2 Möbius → Ch 3 conformal + harmonic → Ch 4 integration / Schwarz → Ch 5 normal families / $\Gamma$ / $\zeta$ → Ch 6 Riemann mapping / Schwarz-Christoffel / Dirichlet → Ch 7 $\wp$ / modular → Ch 8 monodromy / Picard). The Schwarz → normal families → Picard chain in particular must be unbroken after priority-1.
• Pass-W weaving lateral connections: items 16-17 ↔ 06.09.05 + 06.09.06 (plane / RS Mittag-Leffler / Weierstrass parallels); item 25 ↔ 06.06.05 ($\wp$ / theta plane analogue); item 5 ↔ 06.03.03 (Riemann mapping → uniformization); item 26 ↔ 06.06.05 (modular function / theta).

The 5 P1 deepenings close the templated-stub gap (the chapter's largest quality embarrassment). The 15 P1 new units close the Ahlfors-distinctive front-half (sphere, Möbius, elementary functions, Cauchy-Riemann, harmonic, Schwarz, argument principle, normal families) and back-half (gamma, zeta, Weierstrass, Mittag-Leffler, Schwarz-Christoffel, Picard) gaps. P2 (items 21-27) closes Phragmén-Lindelöf / Schwarz reflection / Dirichlet-Perron / $\wp$ / modular function depth — Ahlfors-distinctive, not strict equivalence-blockers. P3 (28-33) is the exercise pack + cross- chapter enrichments. P4 (34-35) optional pointers.

Composite scheduling. Run Batch A first (~10 h) — required so downstream Codex citations point to shipped prose. Items 16-17 retroactively strengthen Forster's 06.09.04-05 (plane prequels). Item 5 deepening feeds Donaldson's 06.03.03 uniformization deepening — composite-batch with Donaldson Phase 1.

Honest scope. Largest single-chapter audit yet — 15 NEW + 5 DEEPEN priority-1, plus 7 NEW + 4 ENRICH P2-3. Work dominated by (a) replacing the v0.5 Strand C/D templated stubs (5 of 6 shipped 06.01.x units) and (b) the entire Ahlfors back-half (gamma / zeta / Weierstrass / Mittag-Leffler / Schwarz-Christoffel / Picard / $\wp$ / modular) which is completely absent from Codex despite being foundational for the existing 06.06-jacobians and 06.09-stein chapters.

Largest Ahlfors-distinctive gap: the back-half Chapter 5-8 block — gamma (item 14), zeta (15), normal families + Picard (13, 19, 20), Schwarz-Christoffel (18). Items 14-15 alone are the single largest topic-level expansion Ahlfors buys Codex (canonical analytic- continuation examples; originator-text foundational).