Synopsis for Further Partial Differential Equations

• Orthogonal functions and eigenfunctions of Sturm–Liouville problems
• General eigenfunction expansions, both discrete and continuous.
• Hankel and Mellin Transforms, with applications.

In weeks 1-8 we will provide an 8 lecture course on eigenfunction expansions and transforms. There will be 4 problem sheets, covering all of the material, for you to try.

1. K. F. Riley, M. P. Hobson & S. J. Bence (2006) Mathematical Methods for Physics and Engineering, 3rd edition, Cambridge University Press.

For discrete eigenfunction expansions, Bessel functions, and orthogonal polynomials, but minimal coverage of Hankel and Mellin transforms.

1. D. G. Duffy (2004) Transform Methods for Solving Partial Differential Equations, 2nd edition, Chapman and Hall/CRC Press.
2. Brian Davies Integral Transforms and Their Applications, 3rd edition 2002, Springer. (Beware of typos . . . )
3. I. N. Sneddon Fourier Transforms, Dover reprint 1995, chapters 1 to 3.
4. L. C. Andrews and B. K. Shivamoggi (1999) Integral Transforms for Engineers, SPIE Press.
5. R. Piessens “The Hankel Transform”
6. J. Bertrand, P. Bertrand and J.P. Ovarlez, “The Mellin Transform” available from http://www.jeanphilippeovarlez.com
7. Two chapters of The Transforms and Applications Handbook, 2nd edition (2000) CRC Press.

Both chapters contain far more material than is necessary, though the expansions in discrete sums from earlier lectures also appears in the Hankel transform chapter as the “finite Hankel transform”.