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RICE UNIVERSITY STUDIES

of functions of complex variables, and of no Otherjit being understood that
functions of a
real variable will expressly identify themselves as such. Thus,
C. Carathéodory, a leading 20th century master of the theory of functions
of a real variable, published in 1918 a large tiTheorie der Funktionen einer
reellen Veranderlichen.*' But, as recently as in 1950, a treatise of his on
complex variables appeared under the title iiFunktionentheorie,"
tout court.
It is true that the treatise was put through publication posthumously. But
this title was undoubtedly so found in the author’s papers, and would have
most likely become Carathéodory’s own, had he lived.

The 19th century also created an in-between topic, namely the topic of
the iiPoisson summation formula” so-called (this appellation is a 20th
century coinage), which seems to fall between real and complex variables,
combining the two with each other and with the analytic theory of arith-
metical forms and of algebraic numbers. Among other things, the Poisson
summation formula generates, by way of “theta relations,” a very remark-
able class of special functions. These are the so-called zeta functions of
number theory, algebraic geometry, and the theory of automorphic functions.
When viewed by themselves, zeta functions appear to be rather isolated
objects of analysis, but the Poisson summation formula as a substantive
background links them with analysis at large.

In a very broad sense, the Poisson summation formula is the key to all
and any “dualities” and “reciprocities” in mathematics, and hence also in
mathematical physics. Dirichlet injected the formula into number theory,
for all time to come, when he demonstrated that, by using the formula, it
becomes “child’s play” to fully derive the reciprocity law for Gaussian
sums, over which Gauss had labored long and hard. The formula is also
the natural setup for dealing with the remainder term in so-called lattice
point problems for euclidean space. Finally, and most gloriously, Erich
Hecke used the formula, and only this formula, for the derivation of the
Riemann-Hecke functional equation for zeta functions over algebraic
number fields.

Regrettably, there is no book as yet dealing with the derivation of various
known zeta functions by means of the Poisson summation formula.

Antiquity. The Greeks, mathematicians and others, did not have the
concept of a (mathematical) function in their thinking. It is not possible to
discern in the body of Greek mathematics something that could be inter-
preted to be an adumbration of the notion of a function
y = /(x) as we
know it today, or, at least, as it is discernible in the mathematics of the 16th
and 17th centuries, say.



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