Alphabeta Math

Complex Analysis

100 pages in 2 parts

Complex Analysis opens with complex differentiability, power series, contour integration and Cauchy's theorems, then follows their consequences into global function theory. Its pages treat zeros and singularities, analytic continuation and monodromy, conformal maps, harmonic and subharmonic functions, potential theory, Hardy spaces and boundary limits. Riemann surfaces bring uniformization, branched maps, divisors, periods and Jacobians; several-variable topics include Hartogs phenomena, analytic hypersurfaces, the Dolbeault complex, pseudoconvexity and the Levi problem. Other branches include quasiconformal mapping and welding, extremal length, Bergman and Szegő kernels, modular forms, and the gamma and zeta functions.

Topology supplies compactness, connectedness and path structure for global arguments; Real Analysis and Measure Theory underpin estimates and boundary limits. Functional Analysis and Partial Differential Equations use contour integration and harmonic methods, while Number Theory uses Cauchy formulas and contour estimates. Complex analysis also supplies tools to Algebraic Geometry, Differential Geometry and Scheme Theory. Together, these methods carry local estimates into global mapping and geometry.

Pathway

The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.

  1. Part 1 · Holomorphic and analytic functions

    3 pages

    Complex differentiability asks for one limit in the plane rather than two along the axes, and the Cauchy-Riemann equations are what that costs: a real total derivative is complex linear exactly when they hold. The identification C≅R2 is made precise: the coordinate map is real-linear, a bijective isometry carrying affine segments and topology, and it transfers real total differentiability; complex path length is defined polygonally, equals Euclidean coordinate length, is finite exactly for bounded-variation coordinates, obeys the piecewise-C1 speed integral and is invariant under monotone reparametrization. A complex power series converges absolutely inside its Cauchy-Hadamard radius and uniformly on smaller discs, so it may be differentiated term by term, and its derivatives recover its coefficients.

  2. Part 2 · Contour integration and Cauchy's theorem

    47 pages · after Part 1

    Contour integration, Goursat and Cauchy turn local holomorphy into analyticity, residues and zero-pole counting; Poisson, Dirichlet, subharmonic, Green and capacity add boundary values and conformal covariance; Fatou, Blaschke and inner-outer factorisation canonicalise Hardy spaces; Bergman–Szegő kernels add reproducing kernels and invariant metrics. Hartogs, pseudoconvexity and Hörmander solve ∂ˉ; Hodge decomposition on compact Kähler manifolds gives harmonic representatives, Lefschetz decomposition, hard Lefschetz and projective Hodge–Riemann polarizations; divisors, Riemann–Roch and Abel–Jacobi follow. Runge-Mittag-Leffler, Weierstrass, Gamma and hypersurfaces approximate; Poisson-Jensen yields Nevanlinna theory; Montel, Schwarz-Pick, Bloch-Schottky-Picard, Riemann mapping, extremal length, quasiconformality, Beltrami solutions, measurable Riemann mapping, welding and removability close the conformal side; monodromy and uniformization globalise; elliptic functions and modular forms classify tori; the zeta page applies Euler products, continuation, functional equation.