THE RISE OF FUNCTIONS*
by Salomon Bochner
Summary. The heart of analysis is the concept of function, and functions
“belong” to analysis, even if, nowadays, they occur everywhere and any-
where, in and out of mathematics, in thought, cognition, even perception.
Functions came into being in “modern” mathematics, that is, in mathe-
matics since the Renaissance. By a rough division into centuries, the 17th
and 18th centuries made various preparations, the 19th century created
functions of one variable, real and complex, and the 20th century has turned
to functions in several variables, real and complex. In the realm of complex
variables, the 20th century has been largely working on themes set by the
19th century, some themes beguiling and pleasant, others harsh and un-
pleasant. Automorphic functions in several variables are intriguing, be-
guiling, and pleasant; singularities of functions or of manifolds of several
variables are harsh, forbidding, and most unpleasant. Which of the two
topics will last longer, and perdure into the 21st century is another ques-
tion; probably the unpleasant one.
For both real and complex variables, the 19th century molded the general
concept of function and also created large classes of special functions, but
there was a difference between the cases of real and complex variables.
In the case of real variables the molding of the general concept of functions
and the creation of special classes of functions proceeded independently
from each other and were performed by different authors, even though
both activities received their motivations from mechanics and physics.
In the case of complex variables, however, the molding of the concept of
function and the creation of special classes of functions were proceeding
in close intimacy and interaction, with Bernhard Riemann being the chief
architect of the dual enterprise.
It is this which made the 19th century into an era of analytic functions
par excellence. Also, in memory of this, the word "Funktionentheorie" in
the title of German books used to indicate, and still indicates, the theme
* Supported in part by a grant from the United States Air Force Office of Scientific
Research.