- COMPLEX ANALYSIS 493
inverses. For example, if q(x) is of degree 5, he posed the problem of solving for ÷
and y in terms of u and í in the equationsThis problem became known as the Jacobi inversion problem. Solving it took a
quarter of a century and led to progress in both complex analysis and algebra.
Jacobi himself gave it a start in connection with elliptic integrals. Although a
nonconstant function that is analytic in the whole plane cannot be doubly periodic
(because its absolute value cannot attain a maximum), a quotient of such func-
tions can be, and Jacobi found the ideal numerators and denominators to use for
expressing the doubly periodic elliptic functions as quotients: theta functions. The
secret of solving the Jacobi inversion problem was to use theta functions in more
than one complex variable, but working out the proper definition of those functions
and the mechanics of applying them to the problem required the genius of Riemann
and Weierstrass. These two giants of nineteenth-century mathematics solved the
problem independently and simultaneously in 1856, but considerable preparatory
work had been done in the meantime by other mathematicians. The importance of
algebraic functions as the basic core of analytic function theory cannot be overem-
phasized. Klein (1926, p. 280) goes so far as to say that Weierstrass' purpose in
life was
to conquer the inversion problem, even for hyperelliptic integrals
of arbitrarily high order, as Jacobi had foresightedly posed it, per-
haps even the problem for general Abelian integrals, using rigor-
ous, methodical work with power series (including series in several
variables).
It was in this way that the topic called the Weierstrass theory
of analytic functions arose as a by-product.1.2. Cauchy. Cauchy's name is associated most especially with one particular
approach to the study of analytic functions of a complex variable, that based on
complex integration. A complex variable is really two variables, as Cauchy was
saying even as late as 1821. But a function is to be given by the same symbols,
whether they denote real or complex numbers. When we integrate and differentiate
a given function, which variable should we use? Cauchy discovered the answer,
as early as 1814, when he first discussed such questions in print. The value of
the function is also a pair of real numbers u 4- iv, and if the derivative is to be
independent of the variable on which it is taken, these must satisfy the equations
we now call the Cauchy-Riemann equations:In that case, as Cauchy saw, if we are integrating u + iv in a purely formal way,
separating real and imaginary parts, over a path from the lower left corner of a rec-
tangle (xo, yo) to its upper right corner (xi, t/i), the same result is obtained whether
the integration proceeds first vertically, then horizontally or first horizontally, then
vertically. As Gauss had noted as early as 1811, Cauchy observed that the function
uídu dv du dv
dx dy' dy dx