Levels 8.1, 8.2 and 8.3 – Real and Complex Analysis
8.1
We start with the actual real Analysis cambridge course. These notes are based on lectures by Rita Teixeira de Costa in the academic 2025-2026. Mistakes are almost surely mine and this is not official nor a recreation of what was in the lectures.
Summary of the course [Expand]
Lectures 1-6 are about sequences and series, and a review of their basic properties. We also cover some tests for series convergence, none of which are particularly surprising.
Lectures 7-10 are about continuous functions, where we formalize "visually obvious" properties such as the intermediate value theorem and the existence of inverse functions under appropriate conditions.
Lectures 10-14 are about formalizingbasic properties of differentiation and giving a few different versions of Taylor's theorem that help us prove taylor series in ways different from what I did in level 4.
Lectures 14-20 are about basic properties of the Riemann integral.
Lectures 21-24 are about formalizing basic properties of power series, and the exponential, logarithm and trigonometric functions.
Basically, due to how much of this content we already covered in levels 3-7, this is more about making it more precise than about learning anything groundbreaking.
We now have an analysis lemmas document with some more lemmas in real analysis that are fundamental to both the vector calculus course and the complex analysis course and even the probability course in a few places, and of which are in level 8. However, this is not technically based on any Cambridge course.
Table of contents [Expand]
Pages 1-6: The inverse function theorem and the implicit function theorem
Pages 7-8: Further properties of the riemann integral
Pages 9-10: Partitions of unity
Page 11: The change of variables theorem
Pages 12-19: Properties of integration on manifolds
We now have the complex analysis course. This is in the second year of Cambridge but I attended lectures a year early so here are my notes. This material is extremely cool and turns out to be important to the probability course in this level. These notes are based on lectures by Ivan Smith in the academic 2025-2026. Mistakes are almost surely mine and this is not official nor a recreation of what was in the lectures.
Summary of the course [Expand]
Lecture 1 introduces basic concepts, such as the Cauchy-Riemann equations. In this course we talk about holomorphic (or complex differentiable) functions.
Lecture 2 talks about conformal mappings and more basic concepts.
Lectures 3 and 4 talk about more basic concepts such as branches and the definition of complex integration
Lecture 5 is where things get interesting. We prove Cauchy's theorem for a triangle and Cauchy's theorem for star-shaped domains, which will imply all the cool results.
In lecture 6 we prove Cauchy's integral formula, probably the most important result of the course since it implies Taylor's theorem and the local maximum principle and Liouville's theorem among other things, and generally constrains holomorphic functions a lot.
In lectures 6-7 we explore these consequences and prove Morera's theorem.
In lectures 7-8 we prove the general Cauchy theorem (ie that holomorphic functions vanish if you integrate them around a loop).
In lectures 8-11 we talk about zeroes, singularities and laurent series.
In lectures 11-12 we formalize Winding number and use it to prove Cauchy's residue theorem.
In lectures 13-14 we use all this to calculate certain integrals.
In lectures 14-16 we go back to theory, doing stuff like the argument principle, proving the open mapping theorem and the riemann mapping theorem as the final theorems of the course.