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Rationale

I began to teach the topics that are included in "Modern Mathematical Methods for Physicists and Engineers" because I wanted to give my students a set of skills that they could truly use as professionals, regardless of their career choices in physics, or modern electrical and telecommunications engineering, or both. I found that, in order to teach the topics in linear algebra, numerical computation, Hilbert space and group theory that I believe are essential, I had to discard the entire traditional sequence of topics in courses on mathematical methods of physics or advanced engineering mathematics and start over.

Topics

The traditional sequence of topics is usually vector analysis, vector calculus, functions of a complex variable, and finally a smattering of functional analysis in the form of a quick overview of the properties of Hilbert space. While vector calculus and functions of a complex variable are undeniably useful, they are well covered in many other books.

Students who are beginning graduate study in electrical and telecommunications engineering need finite fields (for coding theory), linear algebra, and normed vector spaces (for communications theory). Students who are beginning graduate study in physics need group theory, including group representations, for solid-state and elementary-particle physics. Both sets of students need an understanding of the strengths and limitations of computation, which is truly one of the most useful modern mathematical methods of both physics and engineering.

I believe that group theory is almost as important for electrical and telecommunications engineering as it is for physics. As the basic structures from which fields and vector spaces are constructed, groups play a fundamental role in modern engineering. Moreover, the properties of linear, time-shift-invariant systems, which are as basic to signal processing as weightless rigid rods are to elementary mechanics, are the direct result of invariance under the group of translations on the real line. Physicists recognize translational invariance as the origin of the conservation of momentum; electrical engineers recognize translational invariance as the guarantor of the existence of a transfer function for a linear system. Both sets of students need to understand the intellectual underpinnings of their disciplines.

Special features

In traditional approaches, the special functions of mathematical physics appear as solutions of certain obscure differential equations, which, the student is assured, occur frequently. Given that computation is fast and cheap, students are entitled to ask why they need lots of theory to solve differential equations that can be solved numerically (and for which the special-function solutions must be evaluated numerically in any case).

An answer is that students need theory for the purpose of understanding, not for obtaining a numerical answer. Understanding flows most naturally from simple geometrical arguments.

This may be the first textbook (as opposed to a monograph) in which the special functions of mathematical physics are considered as matrix elements of irreducible representations of easily visualized symmetry groups. For example, the Bessel functions of the first kind turn out to be closely related to the Euclidean group − the group of rigid rotations and translations − in two dimensions. The functions Jm can be defined as the Fourier coefficients of the plane wave eik·r, which carries the unitary irreducible representations of the translation subgroup. The reason why one looks for the Fourier series − an expansion in the functions eimφ − is that the latter functions carry the unitary irreducible representations of the subgroup of rotations.

The Bessel functions have raising and lowering operators, which are closely related to recurrence relations, much as for the harmonic-oscillator or angular-momentum eigenfunctions. The statement that a raising operator followed by a lowering operator takes a Bessel function back to a multiple of itself is just Bessel's differential equation. The infinite series for Jm comes from expressing the Taylor series for f(ra) in terms of the Bessel raising and lowering operators, where f(ra) is the product Jm(ka)eimα and α is the polar angle of the vector a. (The derivation of the series is much simpler than it sounds. :)

The realization that the special functions are matrix elements of irreducible representations of various groups is certainly not mine. Credit for this deep insight goes to my dissertation advisor's dissertation advisor, Eugene P. Wigner, a Nobel laureate in physics, and N. Ya. Vilenkin. I simply had the good fortune to sit at Wigner's feet while I was a graduate student.

Topics left out

Some topics on which I prepared material for chapters did not make it into the book, because of time and length (it's already 766 pages!). These topics include:

  • Measure theory, which is important in probability theory and the theory of random processes
  • The Legendre polynomials, spherical harmonics and spherical Bessel functions, which are important in antenna theory and the quantum theory of angular momentum
  • The contraction mapping theorem, which is the basis of important iterative techniques such as the Gauss-Seidel method of solving a large system of linear equations
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