Complex analysis

Calculus of functions of a complex variable — described in the first source the spoke held on it as "the calculus of the plane."

This page was written from a second course with no first course held, and that stopped being true on 2026-08-13. baggett-analysis-single-variable supplies most of the missing layer. The two sources do not quite meet, and the gap between them is narrower and more specific than the one it replaced.

The first course, and the gap that remains

Carroll's preface gives the settled first-course syllabus, and its settledness is the point: analytic functions, contour integrals, Cauchy's Theorem for a disk, the Cauchy Integral Formula, Taylor's Theorem, then Liouville's Theorem, the Fundamental Theorem of Algebra, the Maximum Modulus Theorem, the Residue Theorem, the Argument Principle and Rouché's Theorem. He reports "general agreement" on that list across authors.

baggett-analysis-single-variable covers it as far as Cauchy's Theorem, the Cauchy Integral Formula, Taylor and Laurent expansions, Liouville, the Fundamental Theorem of Algebra, the Maximum Modulus Theorem, the Open Mapping Theorem, isolated singularities and the Residue Theorem — all in one chapter, which he titles "The Fundamental Theorem of Analysis" after Cauchy's Theorem.

Still unheld: the Argument Principle and Rouché's Theorem, the last two items on Carroll's list. Also absent from Baggett is conformal mapping, which is not on the first-course list at all but is where Carroll's book begins — so the corpus holds the two ends of the subject with a thin seam between them rather than an overlap.

Baggett reaches it from real analysis rather than beside it. His Chapter VII deduces Cauchy's Theorem from Green's Theorem, proved the chapter before over plane regions — and his whole book refuses to separate the real and complex developments in the first place (see real-analysis). Carroll's is a course in complex analysis; Baggett's complex analysis is the last third of a course in analysis. That the same theorems arrive either way is worth more than either route on its own.

Why it is a distinct subject and not a chapter of analysis

Two arguments from the source, both worth keeping.

It explains real facts that the real line cannot. Why does the series for 1/(1+x²) have radius of convergence 1 when the function is perfectly well behaved on all of ℝ? Because the real function is the restriction of a complex one with poles at ±i. The complex plane is where the answer lives, which is a claim about necessity rather than convenience.

It is where analysis meets geometry. Carroll places the subject "primarily within the sphere of analysis yet with strong overlap with geometry," and locates the overlap in conformal mapping — analytic one-to-one maps between plane domains. The consequence his whole book builds toward: every simply connected domain carries a natural hyperbolic geometry, invariant under its own conformal self-maps.

Geometric function theory

The geometric branch of the subject, and the corner this corpus actually holds (carroll-geometric-function-theory): the Riemann sphere and the spherical metric, the hyperbolic disk, Schwarz and Schwarz–Pick, normal families and value distribution up to the Great Picard Theorem, the Riemann Mapping Theorem, univalent functions and the Koebe ¼-Theorem, and the Uniformisation Theorem for planar domains.

Where it connects

Carroll names harmonic analysis, differential geometry, algebraic geometry and analytic number theory as fields that assume it, and fluid dynamics, signal processing, control theory and quantum mechanics as applications, with conformal mapping used to solve boundary-value problems.

Two of those touch this hub directly. fourier-analysis is done with complex exponentials throughout, and thibos-fourier-analysis uses phasors without the plane's theory behind them. real-analysis is the sibling development on ℝ, and the contrast is instructive: analyticity in the complex sense is enormously stronger than differentiability in the real one, which is why results like Liouville's have no real analogue.

Open

~~The first course.~~ Mostly closed 2026-08-13 by baggett-analysis-single-variable. What is left of it is narrow: the Argument Principle and Rouché's Theorem, plus conformal mapping as the bridge to the second course.
Curvature. Named by the source as deliberately omitted (§1.4), with Ahlfors and Krantz cited as where it is done. The spherical metric has constant positive curvature and the hyperbolic metric constant negative, and nothing here develops that.
Riemann surfaces, beyond the sphere. Same omission, same source.

carroll-geometric-function-theory · baggett-analysis-single-variable · real-analysis · fourier-analysis · mathematics · tom-carroll · lawrence-baggett