Mathematics for Physicists

Philippe Dennery, André Krzywicki

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486691930
• eBook ISBN: 9780486157122
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

"A fine example of how to present 'classical' physical mathematics." — American Scientist
Written for advanced undergraduate and graduate students, this volume provides a thorough background in the mathematics needed to understand today's more advanced topics in physics and engineering. Without sacrificing rigor, the authors develop the theoretical material at length, in a highly readable, and, wherever possible, in an intuitive manner. Each abstract idea is accompanied by a very simple, concrete example, showing the student that the abstraction is merely a generalization from easily understood specific cases. The notation used is always that of physicists. The more specialized subjects, treated as simply as possible, appear in small print; thus, it is easy to omit them entirely or to assign them to the more ambitious student.
Among the topics covered are the theory of analytic functions, linear vector spaces and linear operators, orthogonal expansions (including Fourier series and transforms), theory of distributions, ordinary and partial differential equations and special functions: series solutions, Green's functions, eigenvalue problems, integral representations.
"An outstandingly complete collection of mathematical material of wide application in physics . . . invaluable to the reader intent on increasing his knowledge of the mathematical theories and techniques underlying physics." — Applied Optics

CHAPTER I

THE THEORY OF ANALYTIC FUNCTIONS

1 · ELEMENTARY NOTIONS OF SET THEORY AND ANALYSIS

1.1 Sets

The notion of a set is basic to all of modern mathematics. We shall mean by set a collection of objects, hereafter called elements of the set. For example, the integers 1, 2, 3, · · ·, 98, 99, 100 form a set of 100 elements. Another example of a set is given by the collection of all points on a line segment; here the number of elements is clearly infinite.

As in the case of other fundamental notions of mathematics (for instance, that of a geometrical point), it is impossible to give a truly rigorous definition of a set. We simply do not have more basic notions at our disposal. Thus, we stated that a set is a “collection of objects,” but of course we would be very embarrassed if we were asked to clarify the meaning of the word “collection.”

The standard way to circumvent the difficulty of defining fundamental mathematical objects is to formulate a certain number of axioms, which are the “rules of the game,” and which form the basis of a deductive theory. The axioms are fashioned upon the intuitive properties of very familiar objects, such as the integers or the real numbers, but once these axioms have been adopted, we need no longer appeal to our intuition. In other words, when the “rules” have been specified, the question of knowing exactly what these objects represent is no longer relevant to the construction of a rigorous theory.

It is possible to develop a rigorous theory of sets based on an axiomatic formulation, but this is completely outside the scope of this book. However, since the theory of sets is now involved in almost all branches of mathematics the use, albeit very limited, of certain concepts and notations of this theory will be very useful to us. It will be quite sufficient for the reader to understand the notion of a set in its most intuitive sense.

1.2 Some Notations of Set Theory

We shall usually denote sets by capital letters; e.g., A, B. Sometimes, however, other symbols will also be used. For instance, (a,b) will denote the set of real numbers satisfying the inequality

a < x < b (1.1)

The symbol

will frequently be used. It is an abbreviation for “belongs to.”

For example, the real number x satisfying the inequalit...

Table of contents

• Cover
• Title Page
• Copyright
• Contents
• Preface
• Chapter I The Theory of Analytic Functions
• Chapter II Linear Vector Spaces
• Chapter III Function Space, Orthogonal Polynomials, and Fourier Analysis
• Chapter IV Differential Equations
• Bibliography
• Index