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Chapter 4

Fourier Series

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4.1 Functions, vibrating strings, and the wave equation.

This entire chapter is devoted to those trigonometric series satisfying certain conditions that go under the general title of Fourier series—named after the French mathematician Joseph Fourier (1768–1830)—but the story of these series begins well before Fourier’s birth.1 And, as you must suspect by now, Euler’s formula plays a prominent role in that story. The prelude to the tale begins with a fundamental question: what is a function?

Modern analysts answer that question ...

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