The History of Mathematics: A Brief Course

502 17. REAL AND COMPLEX ANALYSIS

if r and è are regarded as polar coordinates, while e~ay cos(ax) and e~ay sin(ax)

are harmonic if ÷ and y are regarded as rectangular coordinates. The factor used

to ensure convergence was providing harmonic functions, at no extra cost.

General trigonometric series. The study of trigonometric functions advanced real

analysis once again in 1854, when Riemann was required to give a lecture to qual-

ify for the position of Privatdocent (roughly what would be an assistant professor

nowadays). As the rules required, he was to propose three topics and the faculty

would choose the one he lectured on. One of the three, based on conversations he

had had with Dirichlet over the preceding year, was the representation of functions

by trigonometric series.^11 Dirichlet was no doubt hoping for more progress toward

necessary and sufficient conditions for convergence of a Fourier series, the topic he

had begun so promisingly a quarter-century earlier. Riemann concentrated on one

question in particular: Can a function be represented by more than one trigono-

metric series? That is, can two trigonometric series with different coefficients have

the same sum at every point? In the course of his study, Riemann was driven to

examine the fundamental concept of integration. Cauchy had defined the integral

as Í becomes large, where á = x 0 < x\ < · · • < i„_i < xn = b. Riemann
refined the definition slightly, allowing f(xn) to be replaced by /(x) for any x
between xn-i and xn. The resulting integral is known as the Riemann integral
today. Riemann sought necessary and sufficient conditions for such an integral to
exist. The condition that he formulated led ultimately to the concept of a set of
measure zero,^12 half a century later: For each ó > 0 the total length of the intervals
on which the function f(x) oscillates by more than ó must become arbitrarily small
if the partition is sufficiently fine.

2.2. Completeness of the real numbers. The concept now known as complete-
ness of the real numbers is associated with the Cauchy convergence criterion, which
asserts that a sequence of real numbers {áç}^=1 converges to some real number á
if it is a Cauchy sequence; that is, for every å > 0 there is an index ç such that
Wn — ak\ < å for all fc > n. This condition was stated somewhat loosely by Cauchy
in his Cours d'analyse, published in the mid-1820s, and the proof given there was
also somewhat loose. The same criterion had been stated, and for sequences of
functions rather than sequences of numbers, a decade earlier by Bolzano.

(^11) As the reader will recall from Chapter 12, this topic was not the one Riemann did lecture on.
Gauss preferred the topic of foundations of geometry, and so Riemann's paper on trigonometric
series was not published until 1867, after his death.
(^12) A set of points on the line has measure zero if for every å > 0 it can be covered by a sequence
of intervals (afc,bfc) whose total length is less than e.
as the number approximated by the sums
Í