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Department of Physics

# PHYS3591 Mathematics Workshop (2010/11)

Details of the module's prerequisites, learning outcomes, assessment and contact hours are given in the official module description in the Faculty Handbook - follow the link above.  A detailed description of the module's content, together with book lists, is given below.  For an explanation of the library's categorisation system see http://www.dur.ac.uk/physics/students/library/.

## Content

### Infinite Dimensional Vector Spaces

9 two-hour workshops in Michaelmas Term

Syllabus: Vector Spaces and Hilbert spaces; Linear operators; Matrices; Eigenvalue Problems; Diagonalisation of Matrices; Co-ordinate Transformations; Tensor Calculus.

### Complex Analysis

Prof S.J. Clark

9 two-hour workshops in Michaelmas Term

Syllabus: Analytic functions: functions of complex variable; functions differentiable in the complex sense; Cauchy-Riemann conditions; singularities; multiple-valued functions; branch points and cuts; complex integration and Cauchy's theorem; Taylor and Laurent series; poles and residues; residue theorem and definite integrals; residue theorem and series summation.

### Calculus of Variations and Infinite Series

Dr C.J. Maxwell

9 two-hour workshops in Epiphany Term

Syllabus: Calculus of Variations: Euler-Lagrange equations; Classic variational problems: Brachystochrone; Lagrange Multipliers; Isoperimetric problem. Infinite Series: Convergence criteria; Familiar series; Transformation of series; Taylor series for analytic functions; Asymptotic series. Evaluation of Integrals: Change of variables, Gaussian and related integrals, Gamma Function, miscellaneous methods and tricks.

### Integral Transforms

Prof S.J. Clark

9 two-hour workshops in Epiphany Term

Syllabus: Definition of an integral transform. Fourier series. Fourier transforms; derivation from Fourier series, basic properties and applications, including the convolution theorem, Parseval's relation, and the Wiener-Khinchin theorem. Momentum representation in quantum mechanics. Linear response, causality, and the Hilbert transform. Sampling theorem. Discrete Fourier transform. Laplace transform; relation to Fourier transform, inverse transform and Bromwich integral, basic properties and applications.

Textbooks:

## Teaching methods

Workshops: 2 two-hour workshops per week

The format of the workshops will be a combination of lectures and problem-solving classes. The open-book written examinations will be held in the first week of Epiphany and Easter Terms; the exact dates will be announced at the start of the academic year.