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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 1.3 The General Solution

How did Fourier discover that in order to expand f(x) = 1, the coefficient of should should be ? He gave several several different derivations, but they all amounted to what has become the standard procedure for finding the coefficients in a Fourier series. To keep life simple, we will restrict our attention to even functions which can be expressed only in terms of cosines, f(x) = f(–x). In chapter 6, we will look at the case of Fourier series for more general functions.

We begin with the assumption that our function actually can be written as cosine series, though it may require infinitely many terms. We begin with the equation

(1.3.1)

where the coefficients exist, we just do not know what they are.

There is a nice trick for finding them. We observe that

(1.3.2)

Fourier now uses equation (1.3.2) to peel off the coefficients one at a time:

(1.3.3)

It is now possible to calculate the coefficients for the solution when f(x) = 1. When –1 < x < 1, we have

(1.3.4)

There is one particularly nice consequence of equation (1.3.4). If we set x = 0, then all the cosines take on the value 1. This implies that

(1.3.5)

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