Alphabeta Math

Complex Analysis

54 pages in 2 parts

Complex analysis begins where a single limit in the plane replaces two limits along the axes. Complex differentiability is defined directly, and the Cauchy- Riemann equations are what it costs: a real total derivative is complex linear exactly when they hold, which is the bridge between this subject and multivariable calculus. Complex power series come next, converging absolutely inside the Cauchy-Hadamard radius and uniformly on smaller discs, so a series may be differentiated term by term and its derivatives recover its coefficients, which is what an analytic function is. Integration is then built along paths: the complex line integral is a Riemann-Stieltjes integral along a rectifiable curve, with the ML estimate bounding it by length times supremum. Goursat's theorem gives a vanishing integral over a triangle with no hypothesis on the derivative beyond its existence, and on a convex domain that produces a primitive and so Cauchy's theorem.

The tracks scaffolded above it reserve this material by name. The analytic block of number theory needs the gamma and zeta functions, their continuation and their functional equations, which continue directly from the power series and contour integration here. Functional analysis reserves the power series, contour integration and Cauchy's theorem for its holomorphic functional calculus and its spectral theory. The partial differential equations track reserves contour integration, and the planar harmonic dictionary it uses is the two-dimensional case of this subject.

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

    2 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. 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

    25 pages · after Part 1

    Contour integrals, Goursat, and Cauchy's formula turn local holomorphy into analyticity, derivative estimates, residues, and zero-pole counting. Poisson, Dirichlet, Hartogs, pseudoconvexity, Runge-Mittag-Leffler, Gamma, and Weierstrass extend that control to boundary values, approximation, principal parts, growth, and special functions, while Montel, Schwarz-Pick, Bloch, Schottky, Picard, and Riemann mapping complete the conformal side. Simply connected domains and analytic continuation globalize local germs through periods, primitives, harmonic conjugates, monodromy, and Riemann surfaces. The zeta page then applies the same machinery to a central Dirichlet series: Euler products hold on Re⁡s>1, eta and theta-Mellin formulas continue it meromorphically, and the completed Λ and ξ package the functional equation, zero symmetries, Hadamard product, and special values.