496 17. REAL AND COMPLEX ANALYSIS2. Real analysis
In complex analysis attention is restricted from the outset to functions that have
a complex derivative. That very strong assumption automatically ensures that the
functions studied will have convergent Taylor series. If only mathematical physics
could manage with just such smooth functions, the abstruse concepts that fill up
courses in real analysis would not be needed. But the physical world is full of bound-
aries, where the density of matter is discontinuous, temperatures undergo abrupt
changes, light rays reflect and refract, and vibrating membranes are clamped. For
these situations the imaginary part of the variable, which often has no physical
interpretation anyway, might as well be dropped, since its only mathematical role
was to complete the analytic function. From that point on, analysis proceeds on
the basis of real variables only. Real analysis, which represents another extension
of calculus, has to deal with much more general and "rough" functions. All of the
logical difficulties about calculus poured into this area of analysis, including the
important questions of convergence of series, existence of maxima and minima, al-
lowable ways of defining functions, continuity, and the meaning of integration. As
a result, real analysis is so much less unified than complex analysis that it hardly
appears to be a single subject. Its basic theorems do not follow from one another
in any canonical order, and their proofs tend to be a bag of special tricks, rarely
remembered for long except by professors who lecture on the subject constantly.
The free range of intuition suffered only minor checks in complex analysis. In
that subject, what one wanted to believe very often turned out to be true. But
real analysis almost seemed to be trapped in a hall of mirrors at times, as it strug-
gled to gain the freedom to operate while avoiding paradoxes and contradictions.
The generality of operations allowed in real analysis has fluctuated considerably
over the centuries. While Descartes had imposed rather strict criteria for allowable
curves (functions), Daniel Bernoulli attempted to represent very arbitrary func-
tions as trigonometric series, and the mathematical physicist Andre-Marie Ampere
(1775-1836) attempted to prove that a continuous function (in the modern sense,
but influenced by preconceptions based on the earlier sense) would have a deriva-
tive at most points. The critique of this proof was followed by several decades of
backtracking, as more and more exceptions were found for operations with series
and integrals that appeared to be formally all right. Eventually, when a level of
rigor was reached that eradicated the known paradoxes, the time came to reach for
more generality. Georg Cantor's set theory played a large role in this increasing
generality, while developing paradoxes of its own. In the twentieth century, the
theories of generalized functions and distributions restored some of the earlier free-
dom by inventing a new object to represent the derivative of functions that have
no derivative in the ordinary sense.
2.1. Fourier series, functions, and integrals. There is a symmetry in the
development of real and complex analysis. Broadly speaking, both arose from
differential equations, and complex analysis grew out of power series, while real
analysis grew out of trigonometric series. These two techniques, closely connected
with each other through the relation zn = rn(cosn6 + ismn0), led down divergent
paths that nevertheless crossed frequently in their meanderings. The real and
complex viewpoints in analysis began to diverge with the study of the vibrating
string problem in the 1740s by d'Alembert, Euler, and Daniel Bernoulli.