- BRANCHES AND ROOTS OF THE CALCULUS 477
at all. Given that the independent variable is usually time, that property corre-
sponds to physical determinacy: Knowing the full state of a physical quantity for
some interval of time determines its values for all time. Lagrange, in particular,
was a proponent of power series, for which he invented the term analytic function.
However, as we now know, the natural domain of analytic function theory is the
complex numbers. Now in mechanics the independent variable represents time, and
that fact raises an interesting question: Why should time be a complex variable?
How do complex numbers turn out to be relevant to a problem where only real
values of the variables have any physical meaning? To this question the eighteenth-
and nineteenth-century mathematicians gave no answer. Indeed, it does not appear
that they even asked the question very often. Extensive searches of the nineteenth-
century literature by the present author have produced only the following comments
on this interesting question, made by Weierstrass in 1885 (see his Werke, Bd. 3,
S. 24):
It is very remarkable that in a problem of mathematical physics
where one seeks an unknown function of two variables that, in
terms of their physical meaning, can have only real values and is
such that for a particular value of one of the variables the function
must equal a prescribed function of the other, an expression often
results that is an analytic function of the variable and hence also
has a meaning for complex values of the latter.It is indeed very remarkable, but neither Weierstrass nor anyone since seems to
have explained the mystery. But, just as complex numbers were needed to produce
the three real roots of a cubic equation, it may not have seemed strange that the
complex-variable properties of solutions of differential equations are relevant, even
in the study of problems generated by physical considerations involving only real
variables. Time is, however, sometimes represented as a two-dimensional quantity,
in connection with what are known as Gibbs random fields.
3.2. Partial differential equations. In the middle of the eighteenth century
mathematical physicists began to consider problems involving more than one in-
dependent variable. The most famous of these is the vibrating string problem
discussed by Euler, d'Alembert, and Daniel Bernoulli (1700-1782, son of Johann
Bernoulli) during the 1740s and 1750s. This problem led to the one-dimensional
wave equation
d^2 u _ 292 u
dt?=c d^2 ^^1
with the initial condition u(x,Q) = f(x), f^(x,0) = 0, which Bernoulli solved in
the form of an infinite trigonometric series
oo
an sin nx cos net,oo
the an being chosen so that Ó ansinna; = /(÷).^4
n=l
With this problem, partial differential equations arose, leading to new methods
of solution. The developments that grew out of trigonometric-series techniques are
(^4) This solution was criticized by Euler, leading to a debate over the allowable methods of defining
functions and the proper definition of a function.