Complex analysis

Calculus of functions of a complex variable — described in the spoke’s only source on it as “the calculus of the plane.”

This page is unusual in the corpus: it is written from a second course, with no first course held. Everything below about the elementary layer is what that book states as its prerequisite, not what this wiki has read.

The part the corpus does not hold

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.

None of it is sourced here. This is the corpus’s clearest case of a subject entered from the top.

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. The largest hole any subject page here has: the settled, uncontroversial half of the subject is entirely unheld.
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 · real-analysis · fourier-analysis · mathematics · tom-carroll