Harris Hancock and the Elliptic Realm
Harris Hancock of the University of Cinncinnati wrote a book "Lectures
on the Theory of Elliptic Functions" first published in 1910 by Wiley
and later reprinted by Dover Publications in 1958. In the introduction
on page vii he wrote:
Owing to a theorem due to Liouville, we are able to show the real
significance of the one-valued functions of position on the
Riemann surface, viz., they are the general elliptic functions.
Those one-valued functions form a "class of algebraic functions"
or "a closed realm of rationality," since the sum, difference,
product, or quotient of any two such functions is a function of
the realm. This realm of rationality is of the first order,
corresponding to the connectivity of the associated Riemann
surface, the realm of the ordinary rational functions being of
the zero order. The former realm is derived from the latter by
adjoining an algebraic quantity, which quantity defines the
Riemann surface. This latter realm, which we call the "elliptic
realm," includes as special cases the natural realm of all
rational functions, and also the realm of the simply periodic
functions.
In a metaphorical sense, the "elliptic realm" comprises all of the
mathematical developments associated with elliptic phenomena. This
includes elliptic integrals, elliptic functions, elliptic curves,
and some sequences such as my Somos
Polynomials and generalizations. For an introduction, please
see my essay In the Elliptic Realm (Jun 2016)
(text version)
(or previous version Step into the Elliptic Realm).
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Last Updated Jul 02 2019
Michael Somos <michael.somos@gmail.com>
Michael Somos
"http://grail.eecs.csuohio.edu/~somos/"