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Chapter 1: Crises in Mathematics: Fourier's Series


Chapter 2: Infinite Summations

2.1 Avoiding Infinite Series

2.2 The Geometric Series

2.3 Calculating Pi

2.4 The Harmonic Series

2.5 Taylor Series

2.6 Emerging Doubts

Problem with Series

>Vibrating String

Cauchy's Counter-example

2.6 Emerging Doubts (continued)

The Vibrating String Problem

Fourier's 1807 paper on the propagation of heat was seen by Lagrange and the other members of the reviewing committee as another piece in a longstanding controversy within mathematics. This controversy had begun with the mathematical model of the vibrating string.

In 1747, d'Alembert published the differential equation governing the height y above position x at time t of a vibrating string:
(2.6.1)
where c depends on the length, tension, and mass of the string. To solve this equation, we need to know boundary conditions. If the ends of the string at x = 0 and l are fixed, then y(0,t) = y(l,t) = 0. We also need to know the original position of the string: y(x,0).

The situation is very similar to that of heat propagation. If y(x) can be expressed as a linear combination of functions of the form for example

then the solution to equation (2.6.1) is

It is worth noting that as a function of time, each piece of this solution is periodic. The first piece has period 2l/c; the second has period 2l/3c. That means that the first piece has a frequency of c/2l vibrations per unit time; the second has frequency 3c/2l. This explains the overtones or harmonics of a vibrating string.

Daniel Bernoulli suggested in 1753 that the vibrating string might be capable of infinitely many harmonics. The most general initial position should be an infinite sum of the form

(2.6.2)

Euler rejected this possibility. The reason for his rejection is illuminating. The function in equation (2.6.2) is necessarily periodic with period 2l. Bernoulli's solution cannot handle an initial position that is not a periodic function of x.

Euler seems particularly obtuse to the modern mathematician. We only need to describe the initial position between x=0 and x=l. We do not care whether or not the function repeats itself outside this interval. But this misses the point of a basic misunderstanding that was widely shared in the eighteenth century.

For Euler and his contemporaries, a function was an expression: a polynomial, a trigonometric function, perhaps one of the more exotic series arising as a solution of a differential equation. As a function of x, it existed as an organic whole for all values of x. This is not to imply that it was necessarily well-defined for all values of x, but the values where it was not well-defined would be part of its intrinsic nature. Euler admitted that one could chop and splice functions. For example, one might want to consider for and for But these were two different functions that had been juxtaposed. To Euler, the shape of a function between 0 and l determined that function everywhere.

Lagrange built on this understanding when he asserted that every function has a power series representation and that the derivative of f at x=a can be defined as the coefficient of (x–a) in the expansion of f(x) in powers of (x–a). In other words, he used the Taylor series for f to define f '(a), f ''(a), f '''(a), … As late as 1816, Charles Babbage (1792–1871), John Herschel (1792–1871), and George Peacock (1791–1858) would champion Lagrange's viewpoint. It implies that the values of a function and all of its derivatives at one point completely determine that function at every value of x.

The most revolutionary thing that Fourier accomplished in 1807 was to assert that both Daniel Bernoulli and Leonhard Euler were right. Any initial position can be expressed as an infinite sum of the form given in equation (2.6.2). Fourier showed how to compute the coefficients. But it is equally true that any function represented by such a trigonometric expansion is periodic. The implication is that the description of a function between 0 and l tells us nothing about the function outside this interval.

To Lagrange especially, but probably also to the other members of the committee that reviewed this manuscript, there had to be a flaw. The easiest way out was to assume a problem with the convergence of Fourier's series. In the succeeding years, Fourier and others demonstrated that there was no problem with the convergence. This forced a critical re-evaluation of what was meant by a function, an infinite series, a derivative. As each object of the structure that is calculus came under scrutiny, it was found to rest on uncertain foundations that needed to be examined and reconstructed. Above all, it was the notion of infinity that was in need of correction. One could no longer return to the bliss of an Archimedes and reject the infinite, that would be to reject a hundred and fifty years of the most rapid scientific progress that the world had seen. From celestial mechanics to vibrating systems to fluid dynamics, the scientific interpretations of the world were built upon calculus and the use of infinite series. One could only proceed and hope that there was firmer ground to be found.

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