- COMPLEX ANALYSIS 495
counted according to order, less the number of sheets in the surface, plus l.^3 The
Riemann surface of w = y/z, for example, has two branch points (0 and oo), each
of order 1, and two sheets, resulting in genus 0. Riemann's geometric approach to
the subject brought out the duality between surfaces and mappings of them, en-
capsulated in a formula known as the Riemann-Roch theorem (after Gustav Roch,
1839-1866). This formula connects the dimension of the space of functions on a
Riemann surface having prescribed zeros and poles with the genus of the surface.
Although it is a simple formula to write down, explaining the meaning of the terms
in it requires considerable space, and so we omit the details.
In 1856 Riemann used his theory to give a very elegant solution of the Jacobi
inversion problem. Since an analytic function must be constant if it has no poles on
a Riemann surface, it was possible to use the periods of the integrals that occur in
the problem to determine the function up to a constant multiple and then to find
quotients of theta functions having the same periods, thereby solving the problem.
1.4. Weierstrass. Of the three founders of analytic function theory, Weierstrass
was the most methodical. He had found his own solution to the Jacobi inversion
problem and submitted it simultaneously with Riemann. When he saw Riemann's
work, he withdrew his own paper and spent many years working out in detail how
the two approaches related to each other. Where Riemann had allowed his geomet-
ric intuition to create castles in the air, so to speak, Weierstrass was determined
to have a firm algebraic foundation. Instead of picturing kinematically a point
wandering from one sheet of a Riemann surface to another, Weierstrass preferred a
static object that he called a Gebilde (structure). His Gebilde was based on the set
of pairs of complex numbers (z,w) satisfying a polynomial equation p{z,w) = 0,
where p(z, w) was an irreducible polynomial in the two variables. These pairs were
supplemented by certain ideal points of the form (2,00), (00, uv), or (00,00) when
one or both of w or æ tended to infinity as the other approached a finite or infi-
nite value. Around all but a finite set of points, it was possible to expand w in
an ordinary Taylor series in nonnegative integer powers of æ — ZQ. For each of the
exceptional points, there would be one or more expansions in fractional or nega-
tive powers of æ - z$, as Puiseux and Laurent had found. These power series were
Weierstrass' basic tool in analytic function theory.
Comparison of the three approaches. At first sight, it appears that Cauchy's ap-
proach, which is simultaneously analytic and geometric, subsumes the work of both
Riemann and Weierstrass. Riemann, to be sure, had a more elegant way of over-
coming the difficulty presented by multivalued functions, but Cauchy and Puiseux
between them came very close to doing something logically equivalent. Weierstrass
begins with the power series and considers only functions that have a power-series
development, whereas Cauchy assumes only that the function is continuously dif-
ferentiable.^4 On the other hand, before you can verify Cauchy's basic assumption
that a function is differentiable, you have to know what the function is. How is
that information to be communicated, if not through some formula like a power
series? Weierstrass saw this point clearly; in 1884 he said, "No matter how you
twist and turn, you cannot avoid using some sort of analytic expressions" (quoted
by Siegmund-Schultze, 1988, p. 253).
(^3) Klein (1926, p. 258) ascribes this definition to Alfred Clebsch (1833-1872).
(^4) It was shown by Edouard Goursat (1858-1936) in 1900 that differentiability implies continuous
differentiability on open subsets of the plane.