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This subject builds on, and extends earlier, related undergraduate subjects with topics that are useful to applied mathematics, mathematical physics and physics students, as well as pure mathematics students interested in applied mathematics and mathematical physics. These topics include:
- Special functions: Spherical harmonics including Legendre polynomials and Bessel functions, including cylindrical, modified and spherical Bessel functions;
- Integral equations: Classification, Fourier and Laplace transform solutions, separable kernels, singular integral equations, Wiener-Hopf equations, and series solutions;
- Further vector analysis: Differential forms, and integrating p-forms;
- Further complex analysis: The Schwarz reflection principle, and Wiener-Hopf in complex variables.
Intended learning outcomes
On completion of this subject, students should:
- Be familiar with the most important special functions of mathematical physics, including Legendre polynomials and Bessel functions, and how they arise in solving the Laplace equation in different coordinate systems using separation of variables.
- Learn how a physical problem formulated as a differential equation and a set of boundary conditions can be recast as an integral equation, and how that may offer a way to solve the problem that is not available in the original formulation.
- Be familiar with differential forms as tools that allow one to solve physical problems with maximal notational simplicity.
- Learn new, fundamental concepts that extend the basic concepts of a first subject in complex analysis to allow for the solution of more sophisticated physical problems.
In addition to skills that are useful in careers in science, engineering, commerce and education, students will develop useful generic skills that include:
- The problem-solving skills of identifying strategies to solve unfamiliar problems;
- The analytic skills of constructing and expressing logical arguments, and of working in abstract, general terms to clarify and improve available solutions;
- The time-management skills of meeting regular deadlines while balancing competing commitments.
Last updated: 2 December 2019