The History of Mathematics: A Brief Course

490 17. REAL AND COMPLEX ANALYSIS

complex variable whose range was much vaster than the materials that led to its
creation.
In his 1748 Introductio, Euler emended the definition of a function, saying that a
function is an analytic expression formed from a variable and constants. The rules
for manipulating symbols were agreed on as long as only finite expressions were
involved. But what did the symbols represent? Euler stated that variables were
allowed to take on negative and imaginary values. Thus, even though the physical
quantities the variables represented were measured as positive rational numbers,
the algebraic and geometric properties of negative, irrational, and complex numbers
could be invoked in the analysis. The extension from finite to infinite expressions
was not long in coming. The extension of the calculus to complex numbers turned
out to have monumental importance.
Lagrange undertook to reformulate the calculus in his treatises Theorie des
functions analytiques (1797) and Lecons sur le calcul des fonctions (1801), basing
it entirely on algebraic principles and stating as a fundamental premise that the
functions to be considered are those that can be expanded in power series (having no
negative or fractional powers of the variable). With this approach the derivatives of
a function need not be defined as ratios of infinitesimals, since they can be defined in
terms of the coefficients of the series that represents the function. Functions having
a power series representation are known nowadays as analytic functions from the
title of Lagrange's work.

1.1. Algebraic integrals. Early steps toward complexification were taken only

on a basis of immediate necessity. As we have already seen, the applications of cal-

culus in solving differential equations made the computation of integrals extremely

important. Where computing the derivative never leads outside the class of ele-

mentary functions and leaves algebraic functions algebraic, trigonometric functions

trigonometric, and exponential functions exponential, integrals are a very different

matter. Algebraic functions often have nonalgebraic integrals, as Leibniz realized

very early. The relation we now write as

where ÷ = 1 — cos á.^1 Eighteenth-century mathematicians were greatly helped in

handling integrals like this by the use of trigonometric functions. It was therefore

natural that they would see the analogy when more complicated integrals came to

be considered. Such problems arose from the study of pendulum motion and the

rotation of solid bodies in physics, but we shall illustrate it with examples from

pure geometry: the rectification of the ellipse and the division of the lemniscate

into equal arcs. For the circle, we know that the corresponding problems lead to

the integral

(^1) The limits of integration that we now use were introduced by Joseph Fourier in the nineteenth
century.
was written by him as