Mathematics Colloquium

Thursday, March 8

Time: 3:35 p.m. — 4:30 p.m.

Location: Science Center Room 323

Tags:  Colloquia, Mathematics, Sciences

Cost:  Free

Speaker: Ian Hogan, Central State University

Host: Paul Eloe

Title: Outputs of Generalized Trigonometric Functions

Abstract: Lehmer proved that, up to a factor of two, sine and cosine of rational multiples of pi are algebraic integers, and computed their degrees. Ivan Niven generalized this result to tangent. Kent State graduate student Michael Henry asked if this result can be generalized to trigonometry on other curves, such as the hyperbola and lemniscate. We produce an infinite family of curves on which the result generalizes, and recall results of Gauss and Abel which are analogues of Lehmer's result for the circle and lemniscate. In showing that Lehmer's result does not generalize directly to the hyperbola, we encounter the powerful Gelfond-Schneider theorem, which gives the transcendence of many numbers. We also show how to modify the premises of Lehmer's theorem to produce Fibonacci numbers from sinh.

Refreshments are available at 3:00 PM in SC 313F.

The department colloquia are held every Thursday (excluding holidays) at 3:35 pm in room SC 323 unless otherwise noted. All are invited to attend. 

Contact Information:

Name:  Paul Eloe