What is it about?
The thought-provoking publication by Stephen Wolfram on the history and the future of special functions [1] has encouraged us to prepare the paper devoted to geometrical applications of the function which has been disregarded recently. The following mentions of the Gudermannian have drawn our attention. So, well before the end of the 1800s, essentially all the special functionswe deal with todaywere established. There were some extras too. Like does anyone know what a Gudermannian is? I remember seeing it in books of tables when I was a kid. It’s named after Christof Gudermann, a student of Gauss’s. And it gives a relation between ordinary trig functions and hyperbolic functions.And it’s also relevant to Mercator projections of maps. But somehow it hasn’t really made it to modern times. Well, in developing Mathematica, we regularly talk about the future of special functions. Which functions will really be useful for the kinds of directions people seem to be taking. Which ones are a huge amount of effort to support, but will go theway of the Gudermannian. To revive the disregarded function, we present the first steps of the inverse Gudermannian in the geometry of the hyperbolic spaces of positive curvature. [1] Wolfram S. The History and Future of Special Functions. Given at the Wolfram Technology Conference 2005 in Champaign, IL as part of the Festschrift for OlegMarichev, in honor of his 60th birthday. Available from: http://www.stephenwolfram.com/publications/history-futurespecial-functions/.
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Why is it important?
We present the first applications of the inverse Gudermannian in the geometry of the hyperbolic spaces of positive curvature. A hyperbolic space H^n of positive curvature can be realised on the ideal domain of the Lobachevskii space L^n. In this paper, the metric dependences for various figures of the plane H^2 are written down by the means of the inverse Gudermannian. The volume formula for a finite light cone of the space H^3 is obtained.
Perspectives
In this paper we have presented the first applications of the inverse Gudermannian in the geometry of the hyperbolic spaces H^n of positive curvature. The geometry of the spaces H^n is at the beginning of its development, and the possibility of its application in the research of atomic processes is extremely promising. We believe that the advance of the spaces H^n geometry will maintain the researchers’ interest to the inverse Gudermannian, arousing feelings of gratitude to the mathematicians of the previous generations who devoted their efforts to the studying of this function.
Lyudmila Romakina
Saratovskij nacional'nyj issledovatel'skij gosudarstvennyj universitet imeni N G Cernysevskogo
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This page is a summary of: The inverse Gudermannian in the hyperbolic geometry, Integral Transforms and Special Functions, February 2018, Taylor & Francis,
DOI: 10.1080/10652469.2018.1441296.
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