Chapter 17. Real and Complex Analysis
In the mid-1960s Walter Rudin (b. 1921), the author of a number of standard grad-
uate textbooks in mathematics, wrote a textbook with the title Real and Complex
Analysis, aimed at showing the considerable unity and overlap between the two sub-
jects. It was necessary to write such a book because the two subjects, while sharing
common roots in the calculus, had developed quite differently. The contrasts be-
tween the two are considerable. Complex analysis considers the smoothest, most
orderly possible functions, those that are analytic, while real analysis allows the
most chaotic imaginable functions. Complex analysis was, to pursue our botanical
analogy, fully a "branch" of calculus, and foundational questions hardly entered
into it. Real analysis had a share in both roots and branches, and it was intimately
involved in the debate over the foundations of calculus.
What caused the two varieties of analysis to become so different? Both are
dealing with functions, and both evolved under the stimulus of the differential
equations of mathematical physics. The central point is the concept of a function.
We have already seen the early definitions of this concept by Leibniz and Johann
Bernoulli. All mathematicians from the seventeenth and eighteenth centuries had
an intuitive picture of a function as a formula or expression in which variables are
connected by rules derived from algebra or geometry. A function was regarded as
continuous if it was given by a single formula throughout its range. If the formula
changed, the function was called "mechanical" by Euler. Although "mechanical"
functions may be continuous in the modern sense, they are not usually analytic.
All the "continuous" functions in the older sense are analytic. They have power-
series expansions, and those power-series expansions are often sufficient to solve
differential equations. As a general signpost indicating where the paths diverge, the
path of power-series expansions and the path of trigonometric-series expansions is
a very good guide. A consequence of the development was that real-variable theory
had to deal with very irregular and "badly behaved" functions. It was therefore in
real analysis that the delicate foundational questions arose.1. Complex analysis
Calculus began with a limited stock of geometry: a few curves and surfaces, all of
which could be described analytically in terms of rational, trigonometric, exponen-
tial, and logarithmic functions of real variables. Soon, however, calculus was used
to formulate problems in mathematical physics as differential equations. To solve
those equations, the preferred technique was integration, but where integration
failed, power series were the technique of first resort. These series automatically
brought with them the potential of allowing the variables to assume complex val-
ues. But then integration and differentiation had to be suitably defined for complex
functions of a complex variable. The result was a theory of analytic functions of a
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