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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 1.5 Term-by-term integration

Term-by-term integration and differentiation, the ability to find the integral or derivative of a sum of functions by integrating each summand, works for a finite sum,

It is not surprising that Fourier would assume that it also works for infinite sums of functions. After all, this lay behind one of the standard methods for finding integrals.

Pressed for a definition of integration, mathematicians of Fourier's time would have replied that it is the inverse process of differentiation: To find the integral of f(x), you find a function whose derivative is f(x). This definition has its limitations: What is the integral of ?

There is no simple function with this derivative, but the integral can be found explicitly by using power series. Using the fact that

and the fact that a power series can be integrated by integrating each summand, we see that

(1.5.2)

Mathematicians knew that as long as you stayed inside the interval of convergence, there was never any problem integrating a power series term-by-term. The worst that could go wrong when differentiating term-by-term was that you might lose convergence at the endpoints. Few mathematicians even considered that switching to an infinite sum of trigonometric functions would create problems. But you did not have to press Fourier's solution very far before you started to uncover real difficulties.

Looking at the graph of

shown in figure (1.3), it is clear that the derivative, f '(x), is zero for all values of x other than odd integers. The derivative is not defined when x is an odd integer. But if we try to differentiate this function by differentiating each summand, we get the series

(1.5.3)

which only converges when x is an even integer.

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