Please take this page in conjunction with the Part III Guide to Courses Applied and Computation Analysis section.

The courses in this group tend to focus on the **application** of results from analysis. This means that a basic understanding of real and complex analysis is essential. Many of the courses have strong connections to PDE, so it is also beneficial to have some familiarity with basic notions from this area. Standard results from elementary linear algebra will be assumed. Some of the courses may use some results and ideas from functional analysis so it can be beneficial to have some experience in this area, although generally speaking it is not mandatory.

### Prerequisite areas

- Elementary Linear Algebra: will be assumed for all courses in this area.
- Real Analysis: for Distribution Theory and Applications.
- Complex Analysis: for most courses in this area.
- Basic ODE and PDE theory: for Numerical Solutions of DEs, (Inverse Problems of Imaging).
- Useful but not essential for most courses: Basic Functional Analysis.
- Numerical Analysis and Optimization

## Real Analysis

Relevant Cambridge undergraduate courses are IA Analysis I, IB Analysis II and IB Metric and Topological Spaces.

### Prerequisites

- Uniform convergence
- Differentiability
- Riemannian integration
- Normed spaces
- Metric spaces
- Topological spaces
- Completeness
- Compactness

To aid your preparation, you should make sure that you are familiar with the material on all the example sheets from IB Analysis II and the first example sheet from IB Metric and Topological Spaces. We generally recommend that you concentrate on the main questions and disregard any "extra", "additional" or "starred" questions on the example sheets.

### Useful books and resources

- A series of blogs by Vicky Neale when she was lecturing Ia Analysis I.
- Lecture notes on Metric and Topological Spaces by Prof. Körner from 2015.
*Principles of mathematical analysis*by Rudin.*Real analysis*by Folland.*A second course in mathematical analysis*by Burkill & Burkill.

## Complex Analysis

Relevant Cambridge undergraduate courses are IB Complex Analysis and IB Complex Methods (see General Resources to find the applied maths example sheets).

### Prerequisites

- Cauchy-Riemann equations
- Analyticity
- Convergence of power series
- Möbius maps
- Cauchy's integral theorem
- Morera's theorem
- Residue calculus
- Jordan's lemma
- Fourier transform and inverse

To aid your preparation, you should make sure that you are familiar with the material on all the example sheets from IB Complex Analysis or Complex Methods. We generally recommend that you concentrate on the main questions and disregard any "extra", "additional" or "starred" questions on the example sheets.

### Useful books and resources

- Lecture notes on Complex Analysis by Prof. Scholl.
*Introduction to complex analysis*by Priestey.*Complex variables*by Ablowitz & Fokas.*Real & complex analysis*by Rudin.

## Basic ODE and PDE theory

Relevant Cambridge undergraduate courses are IA Differential Equations, IB Methods and Part II Partial Differential Equations (see General Resources to find the applied maths example sheets).

### Prerequisites

- Picard type existence and uniqueness results
- Flow maps
- Sturm-Liouville theory
- Standard methods of solution of PDEs (e.g. separation of variables, integral transforms, method of characteristics, Fourier series)

Having seen some famous PDEs such as Laplace's equation, the heat equation, the wave equation would help.

The majority of this material is covered in the IB Methods course. You can check your level and guide your preparation by trying the example sheets from that course. We generally recommend that you concentrate on the main questions and disregard any "extra", "additional" or "starred" questions on the example sheets.

### Useful books and resources

- You can find IB Methods lecture notes in several parts on the DAMTP Examples Sheets page.
*Ordinary differential equations*by Arnold.*Introduction to PDEs*by Folland.*Complex Variables*by Ablowitz & Fokas.

## Basic Functional Analysis

It is helpful, though not necessary, to have familiarity with material from Part II Linear Analysis and Part II Analysis of Functions. For example, it might be helpful to have been exposed to some basic Hilbert space theory, familiarity with Sobolev spaces etc. See also the Basic Functional Analysis section on the Analysis page.

### Useful books and resources

*Essential results of functional analysis*by Zimmer.*Functional analysis*by Lax.*Introduction to PDEs*by Folland.*Functional analysis*by Rudin.*Functional analysis*by Reed and Simon.- Lecture notes on Linear Analysis by G. Taylor, based on lectures by B. J. Green and B. Schlein.
- Lecture notes on Analysis of Functions by C. Warnick.

## Numerical analysis and optimization

Some of the courses in the group have a computational aspect, and it is helpful to have some familiarity with numerical analysis and optimization. Relevant courses are IB Numerical Analysis, IB Optimization, and II Numerical Analysis (see this page for exercise sheets).

### Useful books and resources

A First Course in the Numerical Analysis of Differential Equations, by A. Iserles