#### Prerequisites
* Mathematical Methods in Physics (PHYS2611) OR Analysis in Many Variables II (MATH2031).
#### Corequisites
* Foundations of Physics 3A (PHYS3621).
#### Excluded Combination of Modules
* None.
#### Aims
* This module is designed primarily for students studying Department of Physics or Natural Sciences degree programmes.
* It builds on the Level 2 module Mathematical Methods in Physics (PHYS2611).
* It provides the mathematical tools appropriate to Level 3 physics students necessary to tackle a variety of physical problems.
#### Content
* The syllabus contains:
* Vectors and matrices, Hilbert spaces, linear operators, matrices, eigenvalue problem, diagonalisation of matrices, co-ordinate transformations, tensor calculus.
* Complex Analysis: functions of complex variables, differentiable functions, Cauchy-Riemann conditions, Harmonic functions, multiple valued functions and Riemann surfaces, branch points and cuts, complex integration, Cauchy's theorem, Taylor and Laurent series, poles and residues, residue theorem and definite integrals, residue theorem and series summation.
* Calculus of Variations: Euler–Lagrange equations, classic variational problems, Lagrange multipliers.
* Infinite series and convergence, asymptotic series. Integration, Gaussian and related integrals, gamma function.
* Integral Transforms: Fourier series and transforms, convolution theorem, Parseval's relation, Wiener-Khinchin theorem. Momentum representation in quantum mechanics, Hilbert transform, sampling theorem, Laplace transform, inverse Laplace transform and Bromwich integral.
#### Learning Outcomes
Subject-specific Knowledge:
* Having studied this module students will have knowledge of and an ability to use a range of mathematical methods needed to solve a wide array of physical problems.
Subject-specific Skills:
Key Skills:
#### Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module
* Teaching will be by two-hour workshops which are a mix of lectures and examples classes.
* The lectures provide the means to give a concise, focussed presentation of the subject matter of the module. The lecture material will be explicitly linked to the contents of recommended textbooks for the module, thus making clear where students can begin private study.
* When appropriate, the lectures will also be supported by the distribution of written material, or by information and relevant links online.
* New material is immediately backed up by example classes which give students the chance to develop their theoretical understanding and problem solving skills.
* Students will be able to obtain further help in their studies by approaching their lecturers, during the workshop sessions or at other mutually convenient times.
* Student performance will be summatively assessed through two open-book examinations.
* The example classes provide opportunities for feedback, for students to gauge their progress and for staff to monitor progress throughout the duration of the module.
More information at: https://apps.dur.ac.uk/faculty.handbook/2023/UG/module/PHYS3591
