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Fundamental solutions and Gegenbauer expansions of Helmholtz operators on Riemannian spaces of constant curvature
Published
Author(s)
Howard S. Cohl, Thinh Dang, Dunster Mark
Abstract
We compute closed-form expressions for oscillatory and damped spherically symmetric fundamental solutions of the Helmholtz equation in $d$-dimensional hyperbolic and hyperspherical geometry. We are using the $R$-radius hypersphere and $R$-radius hyperboloid model of hyperbolic geometry. These models represent Riemannian manifolds with positive constant and negative constant sectional curvature respectively. Flat-space limits with their corresponding asymptotic representations, are used to restrict proportionality constants for these fundamental solutions. In order to accomplish this, we summarize and derive new large degree asymptotics for associated Legendre and Ferrers functions of the first and second kind. Furthermore, we prove that our fundamental solutions on the hyperboloid are unique due to their decay at infinity. In geodesic polar coordinates, we derive Gegenbauer expansions for these fundamental solutions for $d\ge 3$, and azimuthal Fourier expansions in two-dimensions.
Citation
Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Cohl, H.
, Dang, T.
and Mark, D.
(2018),
Fundamental solutions and Gegenbauer expansions of Helmholtz operators on Riemannian spaces of constant curvature, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), [online], https://doi.org/10.3842/SIGMA.2018.136
(Accessed October 14, 2025)