Complex analysis §
Complex analysis studies functions that are differentiable in the complex plane. Complex differentiability is far more rigid than real differentiability: a once-differentiable function is automatically infinitely differentiable, and its global behavior is controlled by its values on any small open set.
Brief history §
- 18th century: Euler and d’Alembert manipulate complex quantities formally; the Cauchy–Riemann equations appear.
- 19th century: Cauchy proves his integral theorem and formula; Riemann develops the geometric theory (Riemann surfaces, conformal mapping); Weierstrass builds the power-series approach.
- 20th century: the theory of Riemann surfaces and several complex variables grows out of it; residue techniques become a standard tool across mathematics and physics.
Main branches §
| Page | Key question | Core objects |
|---|---|---|
| Holomorphic functions | What does complex differentiability mean, and what rigidity does it force? | Cauchy–Riemann equations, power series, identity theorem, maximum modulus principle |
| Complex integration | How do contour integrals of holomorphic functions behave, and how do they evaluate real integrals? | Cauchy's integral theorem and formula, Liouville's theorem, residue theorem |
| Conformal mapping | Which maps preserve angles, and how do they solve two-dimensional potential problems? | angle-preserving maps, Möbius transformations, Riemann mapping theorem |
| Analytic continuation and Riemann surfaces | How do we extend functions past their domain and treat multi-valued functions geometrically? | analytic continuation, monodromy, branch points, Riemann surfaces |
Details
How the branches nest. Holomorphic functions is the foundation: it defines the object of study (a function differentiable in the complex plane) and the rigidity theorems (identity theorem, maximum modulus principle) that everything else exploits. Complex integration turns that rigidity into a computational engine: Cauchy’s formula expresses a function’s values and derivatives in terms of a contour integral, and the residue theorem reduces a closed contour integral to a finite sum of local data. Conformal mapping uses the angle-preserving property of non-constant holomorphic maps to transfer problems between domains, and Analytic continuation and Riemann surfaces studies what happens when a function’s domain is pushed past its original boundary, including the multi-valued functions that require a geometric surface to be made single-valued.
Out of scope: several complex variables. The history mentions several complex variables, the theory of holomorphic functions of complex variables. It is deliberately not one of the four branches above: in several variables the geometry of domains changes qualitatively (there is no Riemann mapping theorem for general domains, and the maximum modulus principle takes a different form), so it is a distinct subject. The four branches here cover the one-variable theory only.
Where to go next §
- The parent class, with its sibling sub-areas, is Analysis.
- Residue and contour techniques feed back into the other classes: they evaluate real integrals from Real analysis, invert Laplace transforms in Differential equations, and supply spectral tools in Functional analysis.