# PHYS3591 Mathematics Workshop (2011/12)

## 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

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; 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

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

9 two-hour workshops in Epiphany Term

*Syllabus:* Definition of an integral transform. Fourier transforms; derivation from Fourier series, basic properties and applications, including the convolution theorem, Parseval's relation, and the Wiener-Khinchin theorem. Discrete Fourier transform. Laplace transform; relation to Fourier transform, inverse transform and Bromwich integral, basic properties and applications.

*Textbooks:*

Additional: Mathematical Methods of Physics, J. Matthews and R.L. Walker (Wiley)

Additional: Mathematical Methods for Physicists, G.B. Arfken and H.J. Weber (Harcourt)

Additional: Mathematical Methods for Physics and Engineering, K.F. Riley, M. P. Hobson and S.J. Bence (Cambridge)

## 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.