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Chapter 1: Crises in Mathematics: Fourier's Series

Historical Background

Fitting the Initial Condition

The General Solution

> Obstacles

Term-by-term Integration

Section 1.4 Obstacles

Fourier was quick to realize that equation (1.3.4) is only valid for –1 < x < 1. If we replace x by x+2 in the nth summand, then it changes sign:

It follows that for x between 1 and 3, equation (1.3.4) becomes

(1.4.1)

In general, f(x+2) = –f(x). The function represented by this cosine series has a graph which alternates between –1 and +1 as shown in the figure below.

Figure 1.3: Graph of the function

This is very strange behavior. Equation (1.3.4) seems to be saying that our cosine series is the constant function 1. Equation (1.4.1) implies that our series is not constant. Moreover, to the mathematicians of 1807, figure 1.3 did not look like the graph of a function. Functions were polynomials; roots, powers, and logarithms; trigonometric functions and their inverses; and whatever could be built up by addition, subtraction, multiplication, division, or composition of these functions. Functions had graphs with unbroken curves. Functions had derivatives and Taylor series. Fourier's cosine series flew in the face of everything that was known about the behavior of functions. Something must be dreadfully wrong.

Fourier himself recognized one of the serious problems with his technique. That was his assumption that his function had a representation as a cosine series. As he points out in Note 9 of Section 8 of his paper, if he had begun with the assumption that in the interval –1 < x < 1 the function f(x) = 1 could be represented by a series of the form

then he only would have found every other coefficient. Exactly the same reasoning would have led him to conclude that

(1.4.2)

a statement that is wrong. Fourier found his coefficients by a number of different methods, but all of them ultimately relied on the assumption that the cosine series existed.

In retrospect, there is another flaw in his reasoning. That is his assumption in equation (1.3.3) that he could interchange his summation and his integral:

It would be some years before anyone realized that this exchange, which is perfectly legal when the summation is finite, can lead to errors when the summation is infinite.

The process of finding this cosine series was not where his paper was attacked. It was the cosine series itself that presented problems. In section 2.6 we shall investigate the specific objections to trigonometric series that were raised by Lagrange and others. For now let it suffice to say that these infinite summations cast doubt on what scientists thought they knew about the nature of functions, about continuity, about differentiability and integrability. If Fourier's disturbing series were to be accepted, then all of calculus needed to be rethought.

Lagrange thought he found the flaw in Fourier's work in the question of convergence: whether the summation approaches a single value as more terms are taken. He asserted that the cosine series,

does not have a well-defined value for all x. His reason for believing this was that the series consisting of the absolute values of the coefficients,

grows without limit. In fact, Fourier's cosine expansion of f(x) = 1 does converge for any x, as Fourier demonstrated a few years later. The complete justification of the use of these infinite trigonometric series would have to wait twenty-two years for the work of Peter Gustav Lejeune Dirichlet (1805–1859), a young German who, in 1807 when Fourier deposited his manuscript, was two years old.

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