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Chapter 1: Crises in Mathematics: Fourier's Series

> 1.1 Historical Background

1.2 Fitting the Initial Condition

1.3 The General Solution

1.4 Obstacles

1.5 Term-by-term Integration

Chapter 2: Infinite Summations

Section 1.1 Historical Background

Fourier began his investigations with the problem of describing the flow of heat in a very long and thin rectangular plate or lamina. He considered the situation where the top and bottom are insulated so that there is no heat loss from either face of the plate. The two long sides are held at a constant temperature which he set equal to 0. Heat is applied in some known manner to one of the short sides, and the remaining short side is treated as infinitely far away. This sheet can be represented in the x,w plane by a region bounded below by the x-axis, on the left by x = –1, and on the right by x = 1.

Figure 1: The thin rectangular plate (left) and the plate viewed from above (right).

It has a constant temperature of 0 along the left and right edges so that if z(x,w) represents the temperature at the point (x,w), then

The known temperature distribution along the bottom edge is described as a function of x:

Fourier restricted himself to the case where f is an even function of x: f(–x)=f(x). The first and most important example he considered was that of a constant temperature normalized to

(1.1.1)

The task was to find a stable solution under these constraints.

Trying to apply a constant temperature across the base of this sheet raises one problem: What is the value at x=1, w=0? The temperature along the edge x=1 is 0. On the other hand, the temperature across the bottom where w=0 is 1. Whatever value we try to assign here, there will have to be a discontinuity.

But Joseph Fourier did find a solution, and he did it by looking at situations where the temperature does drop off to zero as x approaches 1 along the bottom edge. What he found is that if the original temperature distribution along the interval [–1,1] is given by the function

(1.1.2)

then each term in this sum will drop off exponentially as we move away from the x axis,

(1.1.3)

It is worth noting that the high frequency terms decay faster than the low frequency terms.

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