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Invited Paper Session Abstracts - Special Functions and Constants in Geometry and Trigonometry

MAA Invited Paper Session

Special Functions and Constants in Geometry and Trigonometry

Please note: all sessions are listed in Eastern Daylight Time (EDT = UTC-4:00)

Thursday, August 3, 8:00 a.m. - 10:50 a.m., Room 118/119

The theme is to explore connections between geometric properties and the functions that model those properties. Trigonometry is the classical example of this and generalizations of trigonometric functions to other settings, such as "squigonometry" in the p-norm, invite new ways to explore the interplay of geometry and analysis. This session will focus on this analytic-geometric lens, inviting talks that illustrate how special functions and constants describe geometric objects, and vice versa.

Bill Wood, University of Northern Iowa

Interesting Squigonometric Series

8:00 a.m. - 8:20 a.m.
Robert D. Poodiack, Norwich University

Squigonometric functions are generalized trigonometric functions that parameterize the curves \(|x|^p + |y|^p=1\). As with the usual trigonometric functions, we would like to be able to approximate the values of squigonometric functions and their inverses. We discuss what Maclaurin series for these functions look like, and the oddities that arise in their common structure and their radii of convergence for \(p > 2\). We also will look at some sums computed using these series.

Fourier Analysis of Squigonometric Functions

8:30 a.m. - 8:50 a.m.
Joseph Fields, Southern Connecticut State University

We view the unit \(p\)-squircles as lying in the complex plane. Fourier coefficients can easily be computed to derive complex exponential series for these shapes, and the real and imaginary parts are of course the squigonometric functions. But! There is an interesting degree of freedom introduced by how the squircles are parameterized. Typical choices are that the parameterizations come from arclength and area of the corresponding squircular sectors -- parameterizations that are identical when \(p=2\) but become distinguishable for other values of \(p\). We also consider the related problem of using the squine and cosquine functions as a basis for analternative version of Fourier analysis. There are intriguing differences in the rates of convergence of Fourier series and these alternatives for particular species of periodic functions.

Analysis over Unit P-circles

9:00 a.m. - 9:20 a.m.
Sunil K. Chebolu, Illinois State University

A unit p-circle is the set of all points in the cartesian plane whose distance from the origin equals 1 in the L_p norm. The generalized trigonometric functions parametrize these unit p-circles like their classical circular counterparts. We will explore the geometry of these p-circles, the properties of generalized trigonometric functions, and their analytic continuation.

Zeta Functions and Sums in the Spirit of Ramanujan

9:30 a.m. - 9:50 a.m.
Patrick MacDonald, New College of Florida

Zeta functions make an appearance across a variety of mathematical disciplines including algebra, analysis and geometry. In this talk I will discuss instances of zeta functions occurring in the work of Ramanujan where they make an appearance in the evaluation of infinite sums involving hyperbolic trig functions. I will show that these sums and their evaluation in terms of special values of zeta functions have a natural analytic interpretation. This interpretation leads to interesting expressions that generalize the equalities with which we begin.

The Fundamental Theorem of Starithmetic

10:00 a.m. - 10:20 a.m.
Travis Kowalski, The South Dakota School of Mines & Technology

A regular star polygon is a self-intersecting equilateral, equiangular polygon, and the study of such stars has a rich history in both plane and sacred geometry. In this talk we consider oriented star polygons, which admit a method for constructing new oriented star polygons from two existing ones. We examine the consequences of this construction and prove a ``fundamental theorem" of sorts: every oriented star polygon can be decomposed into a unique sum of irreducible, laterally independent stars. We conclude with a few applications of "starithmetic", including a result expressing rational numbers as unique sums of integers and positive proper fractions.

'A Tale of Two Catenaries’

10:30 a.m. - 10:50 a.m.
Subhranil De, Indiana University Southeast

This work pertains to a ‘double catenary’ that forms when a closed, ideal chain of length L is draped over two frictionless pins at the same vertical height and separated by horizontal distance D. Each of the two segments hanging at equilibrium under the action of gravity forms a catenary. The question studied is whether a given equilibrium solution is stable. We show that although the trivial solution of the two catenaries being identical is always an equilibrium configuration, it is not always stable. There exists a critical value LC for the length such that the trivial solution becomes unstable for L > LC. In such cases, the system is stable only for catenaries of differing lengths, and we present a method to calculate the two said lengths.