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.

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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.

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. 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.

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

It is an abbreviation for belongs to. For example, the real number x satisfying the inequality 1. The set that contains all elements of A and all elements of B, but counted only once, is called the sum or the union of A and B and is denoted by. The set of all elements common to both A and B is called the intersection or sometimes the product, of A and B and is denoted by.

The set of all elements of A that are not included in B is called the difference between A and B and is denoted by. It is convenient to introduce the notion of an empty set, i. It plays the role of the number 0 in algebra and it is also denoted by 0 in this text. The operations with sets can be visualized with the aid of the diagrams of Fig. To end this subsection, we summarize in Table 1 the meaning of the symbols that have been introduced.

The most straightforward classification of sets consists in distinguishing between finite sets i. A more subtle classification consists in distinguishing among infinite sets between enumerable and nonenumerable ones. Otherwise a set is called nonenumerable. For example, any finite set is enumerable. On the other hand, the set of all points of a line segment is nonenumerable; one says that these points form a continuum.

Other examples of nonenumerable sets are the interior of a closed curve in a plane and the interior of a closed surface in space. In the rest of this section we shall consider only the sets of geometrical points; our discussion will apply equally well to sets of points located on a line, in a plane, or in space. A very important notion is that of a neighborhood of a point. For example, the set of all points in a plane lying in the interior of an arbitrary circle centered at the point p is a neighborhood of p , whereas the interior of an arbitrary sphere centered at p stands for the neighborhood of p in space.

S is called an isolated point of the set if there exists a neighborhood of p which does not contain any other. A set of points in a plane: p 1 is an isolated point, since there exists a circle centered at p 1 and which does not contain any of the points of the set. On the contrary, p 2 is an accumulation point.

The points of the segment ab form a closed or an open set, depending upon whether the end points belong or do not belong to the set.

This is in agreement with the intuitive meaning of the word isolated ; effectively, if p is an isolated point, then every element of S is located at a finite distance from p see Fig. A point, every neighborhood of which contains at least one element of S, which is not identical with the point itself, is called an accumulation point of the set. If not only a given point but also all points of some neighborhood of p belong to S, then p is called an interior point of S.

Every interior point of a set is an accumulation point. The converse is not true. Moreover, an accumulation point of a set need not necessarily belong to the set. For example, consider a set of points on a line located between two points a and b , and suppose that a and b do not belong to the set.

It is obvious that any point of the set is an interior point, and consequently an accumulation point. However, the points a and b are also accumulation points of the set, since points of the set come arbitrarily close to a and b. We can now distinguish between two important classes of point sets: A set is called an open set if all its points are interior points; a set is called a closed set if it contains all its accumulation points. Arbitrary point sets are neither open nor closed.

For example, all the points lying within, but not on, a closed curve in a plane form an open set in the plane. If the points lying on the boundary curve are added to the set, it becomes closed. However, the interior points together with several isolated points in the plane form a set which is neither open nor closed. The set of interior points of a segment ab of a line is an open one-dimensional set.

But if the points a and b are added to the set, we get a closed set see Fig. There is a one-to-one correspondence between points on a line and the real numbers, and similarly one can distinguish between an open interval a , b that does not contain the numbers a and b and a closed interval [ a , b ] that does:. To end this section, let us define what we shall later mean by a region: A region is an open set, any two points of which can be connected by a continuous line that is contained entirely within the set see Fig.

It is assumed that the reader is familiar with complex numbers; nevertheless we shall start with a short summary of their properties and of the notation that will be used throughout this chapter.

The points belonging to either one of the shaded areas, but not lying on the boundary curves, form a region. The points p 1 and p 2 as well as the points p 3 and p 4 can be connected by a continuous curve lying within the shaded areas. When the points p 1 and p 3 are connected by a curve, a part of this curve necessarily lies outside the shaded areas. A complex number z is completely specified by a pair of real numbers x and y.

Manipulations with complex numbers are carried out using the usual rules of arithmetic, remembering, however, that by convention. The real numbers x and y are called, respectively, the real and imaginary parts of z and are denoted by Re z and Im z :.

The numbers x and y may be considered as the Cartesian coordinates of a point in a plane. Thus, any complex number can be represented by a point in a plane, hereafter called the complex plane. They are related to the Cartesian coordinates see Fig.

The concept of the modulus of a complex number is simply a straightforward generalization of the concept of the absolute value of a real number. Each complex number z z , with components x and y , in the complex plane. The reader may easily verify that the rule of addition of complex numbers. Since the sum of the lengths of two sides of a triangle is larger than the length of the third side, one immediately gets the triangle inequalities. In this book a bar over any quantity will always mean that we take the complex conjugate of that quantity.

The following relation is evident. To derive inequalities 1. One says that a function has been defined on a set A if one has associated a number in general, complex with every element p A. A is then called the set of arguments of the function. As a familiar example, one can take A to be the set of real numbers x [a,b], and then associate with every x a real number f x.

Such a function can be represented graphically as a curve in a plane see Fig. A more general definition of a function would be to say that we associate not simply one number but some set F p with every p A. As an example, consider a vector field for example, an electric field in space.

With every point of some region in space, one associates a set of three real numbers, the components of the vector, which change from point to point. Let a function f p be defined in a region R; it is irrelevant to our discussion whether R is a set of points located on a line, in a plane, or in space.

If, however, N is independent of p throughout the set, then we say that the sequence 1. The meaning of the uniform convergence of a sequence of functions can be easily illustrated if one considers the simple case of a real function defined on a line or what is equivalent because of the one-to-one correspondence between points on a line and real numbers on an interval [ a , b ]. There exists an important criterion to decide whether or not a sequence of functions is uniformly convergent.

This criterion is certainly known to the reader from elementary calculus. It is, however, more generally valid and holds also when the function is defined on an arbitrary set. The proof is analogous to the one given in the elementary case. The preceding criterion for the uniform convergence of a sequence of functions reduces to the well-known criterion for the convergence of a sequence of numbers when the functions fn p are constant functions.

Hence, the question of the convergence of an infinite series reduces to the problem of the convergence of a sequence of partial sums. In particular, one says that the infinite sum 1. We shall consider in this chapter those functions whose arguments are complex numbers. Defining a function f z over a set of complex numbers amounts to defining a function over a set of points in a plane, the complex plane, that is a function of two real variables.

Since f z is assumed to take on complex values in general, it can always be written as. It is therefore evident that the theory of functions of a complex variable would reduce trivially to the theory of functions of two real variables if the theory of functions of a complex argument was considered in its whole generality. The theory of analytic functions deals, however, only with a restricted class of functions, namely, those functions that satisfy certain smoothness requirements or, to be specific, that are differentiable.

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## Mathematics for Physicists

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