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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

Chapter 2: Infinite Summations

Section 1.2 Fitting the Initial Condition

The problem with the solution in equation (1.1.3) is that it assumes that the distribution of heat along the bottom edge is given by the formula in equation (1.1.2). Any function that can be written in this way must be continuous and equal to 0 at x = ±1. The constant function f(x) = 1 is continuous, but it is not equal to zero at x = ±1. One possible interpretation is that there simply is no solution when f(x) = 1.

That possibility did not sit well with Fourier. After all, it is possible to apply a constant temeprature to one end of a metal bar. Fourier observed that as we take larger values of n, the number of summands in equation (1.1.2), we get functions that more closely approximate f(x) = 1.

Maybe if we could take infinitely many terms, we could get a function of this form that is exactly f(x) = 1. Fourier was convinced this would work and boldly proclaimed his solution:

(1.2.1)

Here was the crux of the crisis. Infinite sums of trigonometric functions had appeared before. Daniel Bernoulli (1700–1782) proposed such sums in 1753 as solutions to the problem of modeling the vibrating string. They had been summarily dismissed by the greatest mathematician of the time, Leonhard Euler (1707–1783). Perhaps Euler scented the danger they presented to his understanding of calculus. The committee that reviewed Fourier's manuscript: Pierre-Simon Laplace (1749–1827), Joseph Louis Lagrange (1736–1813), Sylvestre François Lacroix (1765–1843), and Gaspard Monge (1746–1818), echoed Euler's dismissal in an unenthusiastic summary written by Simeon Denis Poisson (1781–1840). Lagrange was later to make his objections explicit. Well into the 1820s, Fourier series would remain suspect because they contradicted the established wisdom about the nature of functions.

Fourier did more than suggest that the solution to the heat equation lay in his trigonometric series. He gave a simple and practical means of finding those coefficients, the . In so doing, he produced a vast array of verifiable solutions to specific problems. Bernoulli's proposition could be debated endlessly with little effect for it was only theoretical. Fourier's method could actually be implemented. It could not be rejected without forcing the question of why it seemed to work.

There are problems with Fourier series, but they are subtler than anyone realized in that winter of 1807–08. It was not until the 1850s that Bernhard Riemann (1826–1866) and Karl Weierstrass (1815–1897) would sort out the confusion that had greeted Fourier and clearly delineate the real questions.

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