# MA3B8 Complex Analysis

**Lecturer:** Peter Topping

**Term(s):** Term 1

**Status for Mathematics students:** List A

**Commitment:** 30 one-hour lectures

**Assessment:** 3 hour examination (100%).

**Formal registration prerequisites: **None

**Assumed knowledge: **

- MA244 Analysis III
- MA259 Multivariable Calculus
- MA260 Norms Metrics and Topologies or MA222 Metric Spaces

Please note that MA258 Mathematical Analysis III is NOT equivalent to MA244 Analysis III for the purposes of this course.

**Useful knowledge: **The "assumed knowledge" (and their prerequisites) will be enough.

**Synergies: **This course connects with virtually every other domain in both pure and applied mathematics.

**Leads to: **The following modules have this module listed as **assumed knowledge** or **useful background:**

- MA3D1 Fluid Dynamics
- MA475 Riemann Surfaces
- MA4H9 Modular Forms
- MA4L6 Analytic Number Theory
- MA426 Elliptic Curves
- MA4L7 Algebraic Curves
- MA448 Hyperbolic Geometry
- MA4M7 Complex Dynamics

**Content**: The module focuses on the properties of differentiable functions on the complex plane. Unlike real analysis, complex differentiable functions have a large number of amazing properties, and are very rigid objects. Some of these properties have been explored already in second year core. Our goal will be to push the theory further, hopefully revealing a very beautiful classical subject.

In the early part of the module we will see some of the complex analysis topics from MA244 Analysis III, typically in greater depth and/or generality. This includes complex differentiability, the Cauchy-Riemann equations, complex power series, Cauchy's theorem, Taylor's and Liouville's theorem etc. Most of the course will be new topics. We will cover Möbius transformations, the Riemann sphere, winding numbers, generalised versions of Cauchy's theorem, Morera's theorem, zeros of holomorphic functions, the identity theorem, the Schwarz lemma, the classification of isolated singularities, the Weierstrass-Casorati theorem, meromorphic functions, Laurent series, the residue theorem (and applications to integration), Rouché's theorem, the Weierstrass convergence theorem, Hurwitz’s theorem, Montel’s theorem, and the remarkable Riemann mapping theorem (with proof) that ties the whole module together.

**Books**: Please see the Talis-aspire web page of this module for the latest recommended books.