Examples of Heun and Mathieu functions as solutions of wave equations in curved spaces

Abstract

We give examples where the Heun function exists in general relativity. It turns out that while a wave equation written in the background of certain metric yields Mathieu functions as its solutions in four space-time dimensions, the trivial generalization to five dimensions results in the double confluent Heun function. We reduce this solution to the Mathieu function with some transformations.

arXiv comment18 pages, Minor improvements, references added

References in corpus (11)

• On reducing the Heun equation to the hypergeometric equation
• Exact Solutions of Regge-Wheeler Equation and Quasi-Normal Modes of Compact Objects
• Toric Sasaki-Einstein manifolds and Heun equations
• Ince's limits for confluent and double-confluent Heun equations
• On polynomial solutions of Heun equation
• On quasi-normal modes and the AdS_5/CFT_4 correspondence
• Gravitational instantons admit hyper-Kähler structure
• Absorption Cross Section of Scalar Field in Supergravity Background
• Gravitational Instantons from Minimal Surfaces
• Integrability versus separability for the multi-centre metrics
• Gravitational Instantons, Confocal Quadrics and Separability of the Schrödinger and Hamilton-Jacobi equations

Cited by in corpus (25)

• Heun Functions and Some of Their Applications in Physics
• Quasinormal Modes and Greybody Factors of $f(R)$ gravity minimally coupled to a cloud of strings in $2+1$ Dimensions
• Potentials of the Heun class
• Solution of the Dirac equation in the rotating Bertotti-Robinson spacetime
• Quantum Field Theory Applications of Heun Type Functions
• Conformal Invariance and Near-extreme Rotating AdS Black Holes
• Dirac Equation in the Background of the Nutku Helicoid Metric
• Generalization of the three-term recurrence formula and its applications
• Asymptotic behavior of Heun function and its integral formalism
• Symbolic and Numerical Analysis in General Relativity with Open Source Computer Algebra Systems
• Exact helicoidal and catenoidal solutions in Einstein-Maxwell theory
• An Analysis of the Wave Equation for the $U(1{)}^2$ Gauged Supergravity Black Hole
• Wave Equation for the Wu Black Hole
• The analytic solution for the power series expansion of Heun function
• Quantized Black Hole and Heun function
• Singularity Structure and Stability Analysis of the Dirac Equation on the Boundary of the Nutku Helicoid Solution
• Heun and Mathieu functions as solutions of the Dirac equation
• A Newman-Penrose Calculator for Instanton Metrics
• Integrals of Motion in Time-periodic Hamiltonian Systems: The Case of the Mathieu Equation
• Symbolic analysis of second-order ordinary differential equations with polynomial coefficients
• Solutions of a charged scalar field in five-dimensional helicoid solution with electromagnetic field
• Confluent Heun functions and the Coulomb problem for spin 1/2 particle in Minkowski space
• The violation of a uniqueness theorem and an invariant in the application of Poincaré--Perron theorem to Heun's equation
• On the Form of Solutions of Fuchsian differential Equations with n regular singular Points
• Analysis of scalar fields with series convolution