Separable Boundary-value Problems in Physics

Innovative developments in science and technology require a thorough knowledge of applied mathematics, particularly in the field of differential equations and special functions. These are relevant in modeling and computing applications of electromagnetic theory and quantum theory, e.g. in photonics and nanotechnology. The problem of solving partial differential equations remains an important topic that is taught at both the undergraduate and graduate level. The proposed book has a very comprehensive coverage on partial differential equations in a variety of coordinate systems and geometry, and their solutions using the method of separation of variables. The treatment includes complete details on going from the basic theory (including separability conditions not presented in introductory texts) to full implementation for applications. A very good choice of examples is inspired by the authors? research on semiconductor nanostructures and metamaterials and include modern applications like quantum dots. The fluency of the text and the high quality of graphics make the topic easy accessible. The organization of the content by coordinate systems rather than by equation types is unique and offers an easy access. The authors consider recent research results which have led to a much increased pedagogical understanding of not just this topic but of many other related topics in mathematical physics, and which ? like the explicit discussion on differential geometry shows - yet have not been treated in the older texts. To the benefit of the reader, a summary presents a convenient overview on all special functions covered. Homework problems are included as well as numerical algorithms for computing special functions. Thus this book can serve as a reference text for advanced undergraduate students, as a textbook for graduate level courses, and as a self-study book and reference manual for physicists, theoretically oriented engineers and traditional mathematicians. MA4300, PH2300 suitable for graduate level course; could serve as one of two main texts of a partial differential equations course

Morten Willatzen is Head of Research at the Center for Product Innovation of the Mads Clausen Institute at the University of Southern Denmark. Having received his PhD from the Niels Bohr Institute at the University of Copenhagen, he held positions at Aarhus University, Max-Planck-Institute for Solid State Research, Germany, and Senior Scientist at Danfoss A/S, DK. In 2000 he became Associate Professor, in 2004 Full Professor at the University of Southern Denmark. Morten Willatzen's research interests include solid state physics, in particular quantum-confined structures and applications to semiconductor laser amplifiers, flow acoustics, and modelling of thermo-fluid systems.

L. C. Lew Yan Voon is Professor and Chair of the Department of Physics at Wright State University. Educated in Cambridge, England, and Vancouver, Canada, he received his PhD from Worcester Polytechnic Institute, USA, where he held positions until 2004, with a stay at the Max-Planck-Institute for Solid State Research as an Alexander von Humboldt fellow. Dr. Lew Yan Voon was visiting scientist at the Air Force Research Laboratory, Hong Kong University of Science and Technology, Stanford University, and the University of Southern Denmark. Professor Lew Yan Voon received the Balslev Award (Denmark) and the NSF CAREER award. His research interests are in semiconductor theory and mathematical physics and involve the study of band structure theory and applications to nanostructures.

Part I Preliminaries
1. Introduction
2. General Theory
Part II Two-Dimensional Coordinate Systems
3. Rectangular Coordinates
4. Circular Coordinates
5. Elliptic Coordinates
6. Parabolic Coordinates
Part III Three-Dimensional Coordinate Systems
7. Rectangular Coordinates
8. Circular Cylinder Coordinates
9. Elliptic Cylinder Coordinates
10. Parabolic Cylinder Coordinates
11. Spherical Polar Coordinates
12. Prolate Spheroidal Coordinates
13. Oblate Spheroidal Coordinates
14. Parabolic Rotational Coordinates
15. Conical Coordinates
16. Ellipsoidal Coordinates
17. Paraboloidal Coordinates
Part IV Advanced Formulations
18. Differential Geometric Formulations
19. Quantum-mechanical Particle Confined to Neighborhood of Curves
20. Quantum-mechanical Particle Confined to Surfaces of Revolution
21. Boundary Perturbation Theory
Appendices
A Hypergeometric Functions
B Baer Functions
C Bessel Functions
D Lam? Functions
E Legendre Functions
F Mathieu Functions
G Spheroidal Wave Functions
H Weber Functions
I Elliptic Integrals and Functions
Index