Carroll — Geometric Function Theory

Tom Carroll, Geometric Function Theory: A Second Course in Complex Analysis (springer 2024, SUMS, 358pp, ISBN 978-3-031-73726-8, MSC 52A55 / 53A35 / 30-01). Held as a PDF in raw/; read here for its structure, its stated scope and its pedagogy rather than worked through.

It opens complex-analysis in this corpus — a subject the shelf had skipped entirely, running calculus → real-analysis → abstract-algebra with nothing on the complex plane. And it opens it from an unusual end: this is explicitly a second course, and the spoke holds no first one.

What it covers

The through-line is stated on page one: complex analysis “with an emphasis on the geometric viewpoint,” where the link to geometry “derives most of all from conformal mapping.” The payoff the book builds toward is that every simply connected domain carries a natural hyperbolic geometry invariant under its own conformal self-maps.

Ch.	Subject	Landmarks
2	the plane, prepared	Möbius transformations, Schwarz reflection in lines and circles
3	the Riemann sphere	stereographic projection, the spherical metric and its geodesics
4	the hyperbolic disk	Schwarz’s Lemma, disk and half-plane automorphisms, Schwarz–Pick
5	normal families	Arzelà–Ascoli, Montel, Marty, Zalcman, the Great Picard Theorem
6	simply connected domains	homotopy Cauchy, the Riemann Mapping Theorem, conformal mapping of annuli
7	Runge’s Theorem	rational and polynomial approximation; another characterisation of simple connectivity
8	univalent functions	the Area Theorem, Bieberbach’s coefficient estimate, the Koebe ¼-Theorem
9	Carathéodory convergence	convergence of domains; hyperbolic geodesics, after Jørgensen
10	uniformisation	covering spaces, the modular function, Picard again, the Uniformisation Theorem for planar domains

Then a glossary, 54 pages of full solutions to the exercises, references and an index. Chapters 8, 9 and 10 are written to be read independently of each other.

The organising device is the Schwarz–Pick Lemma, which appears in three successive versions — disk (ch. 4), simply connected domain (ch. 6), and the comparison results of ch. 9 — so one theorem is re-proved at widening generality as the machinery arrives. Chapters 3 and 4 deliberately cover similar Riemannian ground twice by opposite routes: on the sphere the metric is built first and its isometries found afterwards; on the disk the isometries are decided in advance, guided by Schwarz’s Lemma, and the metric constructed to fit.

The finding: canon exists for the first course and dissolves after it

The preface states it directly. “There is general agreement on the list of main topics that comprise a first course in complex analysis” — Cauchy’s Theorem, the Integral Formula, Taylor, Liouville, Maximum Modulus, Residues, the Argument Principle, Rouché — and “the general syllabus of a first course in complex analysis is well-established.” Then: “This consensus evaporates when it comes to a second course.”

That is the sharpest statement this corpus holds about how mathematical curricula are made. Everything else here is a first course, where the author inherits a syllabus and argues about rigour, foundations or exercise design within it (keisler-elementary-calculus against ε-δ, judson-abstract-algebra on where to place applications). Carroll has no syllabus to inherit, so his scope decisions are his own and he has to defend them — which is why this book does something none of the others do.

§1.4, “What Is Not in This Book”

A numbered section listing what he left out and where to go instead: no in-depth treatment of conformal invariants beyond the hyperbolic metric (defers to Ahlfors’s Conformal Invariants), no curvature (defers to Ahlfors and to Krantz’s Complex Analysis: The Geometric Viewpoint), no proper treatment of Riemann surfaces except the sphere, and nothing on Denjoy–Wolff or Julia–Wolff–Carathéodory — which he says “would fit naturally in this book.”

He concedes the general point too: “Even if every topic covered here has earned its place, that is not to say that many other topics could lay equal claim to being included.”

An explicit non-goals section is rare in a textbook and it is exactly what this spoke reads texts for. Compare coding-interview-university in ../engineering-education-wiki, which excludes frontend and SQL because the interview does: there the omission is set by an exam, here by a author with no exam behind him, and the difference is that Carroll names his omissions and says where they are treated properly instead.

Level, prerequisites and apparatus

Aimed at final-year undergraduates in the SUMS series, whose stated model is “a one- or two-semester course” with “fully-worked solutions” and a self-study path. Prerequisite: a first course in complex analysis, with Howie’s Complex Analysis (same series) named as sufficient and Stein–Shakarchi recommended alongside.

Two apparatus notes worth recording against the corpus’s pedagogy axis. The exercises come with complete solutions (pp. 295–348), where corral-vector-calculus‘s 420 exercises give answers only for a subset; this is a self-study book by design. And it assumes no computational tool — no Sage as in judson-abstract-algebra, no Matlab as in thibos-fourier-analysis. The subject’s objects are pictures, and the book draws them.

Tier

T1. A published Springer textbook read in the original — held in raw/, so the chapter structure, the preface and §1.4 are quoted from the source rather than from a catalogue page (unlike lal-algebra-1, whose page rests on the publisher’s description). Not free: the corpus’s access pattern breaks again, as it did with larsen-marx-mathematical-statistics and lal-algebra-1.

What this page does not claim. The mathematics has not been verified and the proofs have not been read. This records what the book contains, at what level, and what it says about its own choices.

complex-analysis · tom-carroll · springer · springer-undergraduate-mathematics-series · university-college-cork · real-analysis · mathematics · synthesis