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.
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.
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
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
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
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(r − a) in terms of the Bessel raising and lowering
operators, where f(r − a) is the product
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.
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:
Some of this material may be made available on this site at a future time.
- 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