Separation of Variables and Contractions on Two-Dimensional Hyperboloid

TL;DR

This paper studies how solutions to the Helmholtz equation on a two-dimensional hyperboloid behave under a specific contraction limit, deriving new asymptotic formulas for these basis functions.

Contribution

It introduces analytic contractions for basis functions of the Helmholtz equation on the hyperboloid and derives new asymptotic formulas in the contraction limit.

Findings

Established analytic contractions in the $R oinity$ limit

Derived new asymptotic formulas for basis functions

Provided insights into the behavior of solutions on hyperboloids

Abstract

In this paper analytic contractions have been established in the $R\to\infty$ contraction limit for exactly solvable basis functions of the Helmholtz equation on the two-dimensional two-sheeted hyperboloid. As a consequence we present some new asymptotic formulae.