Transcendental Equations for Nonlinear Optimization in Hyperbolic Space

We present a novel application of transcendental equations for nonlinear distance optimization in hyperbolic space. Through asymptotic approximations using Fourier and Taylor series expansions, we obtain approximations for the transcendental equations with non-zero real values on the boundary λ. The series expansion of the logarithmic form of our equations around two arbitrary points P1 and P2 can be used to find values close to definite coordinates on λ. Applying principles from the Poincaré hyperbolic disk—a non-Euclidean space with constant negative curvature—we construct optimization methods following λ of our transcendental equations.