Complex Analysis

Reference notes on the theory of holomorphic functions — differentiable functions of a complex variable — and the classical edifice built on them: contour integration and residues, analytic continuation, geometric function theory, the factorization and special-function theory, and asymptotics. The aim is a self-contained complex-analysis reference at the level of Ahlfors / Stein–Shakarchi / Conway.

Complex analysis is also the most-used piece of mathematics in the physics tree — the Feynman $i\epsilon$ prescription, loop integrals, Wick rotation, dispersion relations, and the saddle-point (semiclassical) method are all complex-analytic — so one strand of the folder feeds physics, collected in the physics bridge.

These pages are companion material to the Mathematics section. They run parallel to the real analysis spine — assuming its limits, series, and integration — and open with a foundations page that constructs $\mathbb{C}$, its metric and topology, and the complex limit.

Contents

A. Foundations and Cauchy theory

1. The Complex Plane: Field, Metric, and Limits — the field $\mathbb{C}$, its geometry as ${\mathbb{R}}^2$, the topology and completeness of the plane, complex limits, and the Riemann sphere.
2. Holomorphic Functions and the Cauchy–Riemann Equations — complex differentiability, the CR equations, harmonicity, and power series.
3. Cauchy's Theorem and Integral Formula — contour integrals, Cauchy's theorem (with its homology form and winding numbers), the integral formula, and its startling corollaries.

B. Series and residues

4. Laurent Series and Residues — expansions about singularities, the classification of singularities, and the residue theorem.
5. Evaluating Integrals by Residues — real integrals via contours, Jordan's lemma, principal values, series summation, and the $i\epsilon$ prescription.

C. Geometric function theory

6. Conformal Mapping — angle-preserving maps, Möbius transformations, the Riemann mapping theorem, and 2D CFT.
7. Harmonic Functions and Potential Theory — the Poisson integral, the Dirichlet problem, Harnack's inequality, and subharmonic functions.
8. The Schwarz Lemma, Disc Automorphisms, and the Hyperbolic Metric — Schwarz–Pick, Blaschke factors, and the Poincaré metric.
9. Normal Families and the Riemann Mapping Theorem — Montel's theorem, Hurwitz, and the proof of the mapping theorem.

D. Global structure and continuation

10. Analytic Continuation and Dispersion Relations — the identity theorem, monodromy, germs and natural boundaries, Kramers–Kronig relations, and Wick rotation.
11. Riemann Surfaces and Branch Cuts — multivalued functions made single-valued; branch points, sheets, uniformization, and physical thresholds.

E. Entire, meromorphic, and special functions

12. Infinite Products, Factorization, and Mittag-Leffler — Weierstrass and Hadamard factorization, Jensen's formula, order and genus.
13. The Gamma Function — the Euler integral, the Weierstrass product, the functional and reflection formulas, and Stirling.
14. The Riemann Zeta Function and the Prime Number Theorem — the Euler product, the functional equation, the zeros, and the PNT.
15. Value Distribution: The Picard Theorems — the little and great Picard theorems and a Nevanlinna-theory preview.

F. Elliptic and modular functions

16. Elliptic Functions and Modular Forms — doubly-periodic functions, the Weierstrass $\wp$, theta functions, and the modular group.

G. Asymptotics and the physics bridge

17. Saddle Point and Steepest Descent — Laplace's method, stationary phase, and the semiclassical expansion.
18. Why This Matters — The Physics Bridge — poles, residues, cuts, $i\epsilon$, and Wick rotation, collected.

H. Outlook

19. Several Complex Variables (Outlook) — Hartogs' phenomenon, domains of holomorphy, and the failure of the mapping theorem.

Reading order

The core spine runs $$\text{complex-plane}\to \text{holomorphic}\to \text{cauchy-integral}\to \text{laurent-residues}\to \text{evaluating-integrals}\to \text{analytic-continuation},$$ which carries the differential/integral heart of the subject (and everything a QFT loop calculation needs). Three streams branch off it and can be read in any order: geometric function theory (conformal mapping → harmonic functions → Schwarz lemma → normal families, which proves the Riemann mapping theorem); entire and special functions (infinite products → Gamma → zeta, with Picard and elliptic functions off it); and asymptotics (saddle point). The physics bridge collects the dictionary, and several complex variables marks the outer boundary. Prerequisites (limits, series, real integration) come from the analysis spine; in particular Cauchy's theorem is Green's theorem plus the Cauchy–Riemann equations. Readers who only need a specific technique can jump in anywhere.