Introduction to Calculus and Analysis I

Since antiquity the intuitive notions of continuous change, growth, and motion, have challenged scientific minds. Yet, the way to the understanding of continuous variation was opened only in the seventeenth century when modern science emerged and rapidly developed in close conjunction with integral and differential calculus, briefly called calculus, and mathematical analysis.

The fundamental limiting processes of calculus are integration and differentiation. Isolated instances of these processes of calculus were considered even in antiquity (culminating in the work of Archimedes), and with increasing frequency in the sixteenth and seventeenth centuries. However, the systematic development of calculus, started only in the seventeenth century, is ussually credited to the two great pioneers of science, Newton and Leibnitz. The key to this systematic development is the insight that the two processes of differentiation and integration, which had been treated separately, are intimately related by being reciprocal to each other.¹

Although problems of integration are usually of greater importance than those of differentiation, the latter offer less formal difficulty than the former. Therefore it is a natural procedure first to master the art of differentiating the widest possible classes of functions ; then by the fundamental theorem (Section 2.9) the results of differentiation are available for evaluating integrals. In the following sections we shall pursue such applications of the fundamental theorem. To a certain extent we shall make a fresh start and develop techniques of integration systematically on the basis of certain general rules for differentiation.

The representation of a curve by an equation y = f(x) imposes a serious geometrical restriction: A curve so represented must not be intersected at more than one point by any parallel to the y-axis. Usually, this restriction can be overcome by decomposing the curve into portions each representable in the form y = f(x). Thus a circle of radius a about the origin is given by the two functions and defined for -a ≤ x ≤ a. However, for as simple a curve as a parallel to the y-axis this device does not work.

It was a great triumph in the early years of Calculus when Newton and others discovered that many known functions could be expressed as “polynomials of infinite order” or “power series,” with coefficients formed by elegant transparent laws. The geometrical series for 1/(1 − x) or 1/(1 + x ²) valid for the open interval |x| < 1, are prototypes (see Cahpter 1, p. 67).

The task of solving an analytical problem always remains uncompleted. The proof of the existence and of some basic properties of the solution is usually considered satisfactory, but relevant questions always remain to be answered. Thus, when the solution is defined by a limit process, for example by an integral, the problem arises of actually finding approximations to this limit and of estimating the accuracy of these approximations. Not only are such questions of basic importance theoretically but they are also inevitable, if we wish to apply analysis to the description and control of natural phenomena which in principle can be described only in an approximate manner.

The functions represented by power series, or as Lagrange called them, the “analytic functions,” play indeed a central role in analysis. But the class of analytic functions is too restricted in many instances. It was therefore an event of major importance for all of mathematics and for a great variety of applications when Fourier in his “Théorie analytique de la chaleur”¹ observed and illustrated by many examples the fact that convergent trigonometric series of the form with constant coefficients a _v, b _v are capable of representing a wide class of “arbitrary” functions f(x), a class which includes essentially evry function of specific interest, whether defined geometrically by mechanical means, or in any other way: even functions possesing jump discontinuities, or obeying different laws of formation in different intervals, can thus be expressed.