Aims: A two-part course covering an introduction to asymptotics, and an introduction to integral transforms, focusing on their properties and their applications, with proofs to come in later courses (although these may be hinted at by the lecturer). The course covers standard techniques that are of widespread use throughout applied mathematics, physics, and engineering.

Content: We will cover the following topics:

Asymptotics:

• Formal definition of an asymptotic series, with examples (e.g. erf(z)). Discussion of the origins of small parameters (e.g. dimensionless parameters, stability analysis)
• Asymptotics of algebraic equations, with examples (e.g. solutions of nearly-linear quadratic equations)
• Asymptotics of integrals, with examples (e.g. Stirling's formula, computing oscillatory integrals)
• Asymptotics of differential equations, with examples (e.g. boundary layers).

Integral Transforms:

• Definition of an Integral Transform, with examples including Fourier (superposition of musical notes) and Radon (CAT scans) transforms
• Fourier Transforms, and their applications to linear ODEs and PDEs, with examples including waves in a waveguide
• Laplace transforms, and their use in solving initial-value problems for ODEs
• A brief tour of other integral transforms, including Mellin, Z, and Radon transforms

If time permits, we may also touch on Green’s Functions, Discrete Fourier Transforms or Half-Range Fourier Transforms.

Objectives:

By the end of the module, students should be able to:

• Understand the formal definition of asymptotic series and their uses
• Be able to identify both regular and singular perturbations
• Use various techniques to construct asymptotic series
• Be aware of a range of integral transforms and their interpretations
• Be able to calculate Fourier and Laplace transforms and understand their similarities and differences