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Rating: 
Summary: A wonderful book to work through, but not a reference book
Review: This is a great book if you want to learn how to use mathematics to solve applied problems, especially boundary value problems. Mr. Dettmann manages to cover a wide range of topics (see chapter list below) in only 400 pages. He manages to to keep it short because of his clear, concise prose, and because he allows the reader to derive many interesting and important results in the problems. Thus, this is NOT the book for you is you are looking for a reference with recipes. Each chapter builds on material in previous chapters, and for the first time I saw the connections between the various methods that physicists and engineers use. This book is definitely more rigorous than the engineering mathematics courses I have taken. Convergence of series and completeness of eignefunctions are dealt with in detail. The book is written for upper level undergraduates with reasonable backgrounds in calculus, vector analysis, and basic ODEs. It is assumed that the reader is familiar with uniform convergence - a few hours with an advanced calculus book will provide you with everything you need, though. Chapters: 1. Linear Algebra, including sections on systems of ODEs and linear programming. 2. Hilbert Spaces, including a nice section on Fourier series, and the spectral theory of completely continuous linear operators. 3. Calculus of Variations, includes discussions of Hamilton's principle from mechanics, and a lot of discussion of boundary value problems. This is the longest chapter in the book (70 pages!) 4. Boundary Value Problems: separation of variables, Separation of variables is actually introduced in chapter 3. this chapter provides a general discussion of othogonal coordinate systems, Sturm-Liouville problems, series solutions of ODEs, and series solutions of PDEs. 5. Boundary Value Problems: Green's functions, sections deal with a general discussion of the non-homogeneous PDE, Green's functions for ODE's, a really nice outline of the theory of generalized functions, Green's functions in higher dimensions, unbounded domains, and an example from diffraction theory. This chapter will not tell you how to solve problems in Jackson, but it will give you the tools so that you can figure out the solution methods for yourself! 6. Integral Equations, formulating BVPs as integral equations, Hilbert-Schmidt theory, Freedholm theory. 7. Analytic Function theory, I haven't read this chapter, but it looks reasonable. I did notice that there is no discussion of multivalued functions, which limits the examples in Chapter 8. 8. Integral transform methods, Emphasizes the Fourier transform (including the case with complex argument) and solutions of ODEs, PDEs, and integral equations. Also has a section on Laplace transforms. Includes very brief discussions of finite Fourier, Hankel, and Mellin transforms, and the Wiener-Hopf technique.
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