The History of Mathematics: A Brief Course

498 17. REAL AND COMPLEX ANALYSIS

of gravity, Pierre-Simon Laplace (1749-1827) was led to what is now known as

Laplace's equation in three variables. The two-variable version of this equation is

d^2 u d^2 u _

dx^2 dy^2

The operator on the left-hand side of this equation is known as the Laplacian.

Since Laplace's equation can be thought of as the wave equation with velocity c =

v/—T, complex numbers again enter into a physical problem. Recalling d'Alembert's

solution of the wave equation, Laplace suggested that the solutions of his equation

might be sought in the form f(x + é/\/-ú) + - õ\/—ú). Once again a problem

that started out as a real-variable problem led inexorably to the need to study

functions of a complex variable.

The definition of a function. Daniel Bernoulli accepted his father's definition of a
function as "an expression formed in some manner from variables and constants,"
as did Euler and d'Alembert. But those words seemed to have different meanings
for each of them. Daniel Bernoulli thought that his solution met the criterion of
being "an expression formed from variables and constants." His former colleague
in the Russian Academy of Sciences,^6 Euler, saw the matter differently. This time
it was Euler who argued that the concept of function was being used too loosely.
According to him, since the right-hand side of Bernoulli's formula consisted of odd
functions of period 2L, it could represent only an odd function of period 2L. There-
fore, he said, it did not have the generality of the solution he and d'Alembert had
given. Bottazzini (1986, p. 29) expresses the situation very well, saying, "We are
here facing a misunderstanding that reveals one aspect of the contradictions be-
tween the old and new theory of functions, even though they, are both present in
the same man, Euler, the protagonist of this transformation." The difference be-
tween the old and new concepts is seen in the simplest example, the function |x|,
which equals ÷ when ÷ > 0 and -x for ÷ < 0. We have no difficulty thinking of
this function as one function. It appeared otherwise to nineteenth-century mathe-
maticians. Fourier described what he called a "discontinuous function represented
by a definite integral" in 1822: the function

2 r°° cosgx _ (e~x ifx>0,

i/o 1 + q2 Q ~ { ex if ÷ < 0.

Fifty years later Gaston Darboux (1844-1918) gave the modern point of view, that
this function is not truly discontinuous but merely a function expressed by two
different analytic expressions in different parts of its domain.
The change in point of view came about gradually, but an important step
was Cauchy's refinement of the definition in the first chapter of his 1821 Cours
d 'analyse:

When variable quantities are related so that, given the value of

one of them, one can infer those of the others, we normally con-

sider that the quantities are all expressed in terms of one of them,

which is called the independent variable, while the others are called

dependent variables.

Bernoulli had left St. Petersburg in 1733, Euler in 1741.