Classical Topics

Abstract

Prior to the middle of the nineteenth century, mathematicians thought about functions very much as do beginning calculus students today : A function is given by a formula. As an extreme example, Leonhard Euler (1707–1783) addressedone of the great questions of the late eighteenth century—whether an arbitrary set of data for the wave equation (i .e., any function representing the initial position of a vibrating string) has an expansion in terms of sines and cosines—as follows: One possible initial configuration for the string on the interval [0, 2:π] is

$$ \varphi (x) = \left\{ {\begin{array}{*{20}c}{x^2 /\pi if 0 \leqslant x \leqslant \pi ;} \\{2\pi - x if \pi < x \leqslant 2\pi .} \\ \end{array} } \right.$$