494 17. REAL AND COMPLEX ANALYSISl/(x + iy) did not have this property if the rectangle contained the point (0,0).
The difference between the two paths was 2ðß, which Cauchy called the residue.
Over the period from 1825 to 1840, Cauchy developed from this theorem what is
now known as the Cauchy integral theorem, the Cauchy integral formula, Taylor's
theorem, and the calculus of residues. The Cauchy integral theorem states that if
7 is a curve enclosing a region in which f(z) has a derivative thenIf the real and imaginary parts of this integral are written out and compared with
the Cauchy-Riemann equations, this formula becomes a simple consequence of what
is known as Green's theorem (the two-variable version of the divergence theorem),
published in 1828 by George Green (1793 1841) and simultaneously in Russia by
Mikhail Vasilevich Ostrogradskii (1801-1862). When combined with the fact that
the integral of 1/z around a curve that winds once around 0 is 2ðú, this theorem
immediately yields as a consequence the Cauchy integral formula
When generalized, this formula becomes the residue theorem. Also from it, one
can obtain estimates for the size of the derivatives. Finally, by expanding the
denominator as a geometric series in powers of æ — æ÷, where Æ÷ lies inside the
curve 7, one can obtain the Taylor series expansion of f(z). These theorems form
the essential core of modern first courses in complex analysis. This work was
supplemented by a paper of Pierre Laurent (1813-1854), submitted to the Paris
Academy in 1843, in which power series expansions about isolated singularities
(Laurent series) were studied.
Cauchy was aware of the difficulties that arise in the case of multivalued func-
tions and introduced the idea of a boundary curve (ligne d'arret) to prevent a
function from assuming more than one value at a given point. As mentioned in
Chapter 12, his student Puiseux studied the behavior of algebraic functions in the
neighborhood of what we now call branch points, which are points c such that
the function assumes many different values at each point of every neighborhood of
c. Puiseux showed that at a branch point c near which there are ç values of the
function each of the ç values of the function could be expanded in its own series of
powers of a variable u such that it" — x — c. The work of Cauchy, Laurent, Puiseux,
and others thus brought complex analysis into existence as a well-articulated theory
containing important principles and theorems.
1.3. Riemann. The work of Puiseux on algebraic functions of a complex variable
was to be subsumed in two major papers of Riemann. The first of these, his doc-
toral dissertation, contained the concept now known as a Riemann surface. It was
aimed especially at simplifying the study of an algebraic function w(z) satisfying
a polynomial equation P(z, w(z)) =0. In a sense, the Riemann surface revealed
that all the significant information about the function was contained precisely in its
singularities—the way it branched at its branch points. Information about the sur-
face was contained in its genus, defined as half the total number of branch points,
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