Asymptotic behavior of Heun function and its integral formalism

Abstract

The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice systems in statistical mechanics, solution of the Schrodinger equation of quantum mechanics, and addition of three quantum spins. In this paper, applying three term recurrence formula, I consider asymptotic behaviors of Heun function and its integral formalism including all higher terms of A_n's. I will show how the power series expansion of Heun functions can be converted to closed-form integrals for all cases of infinite series and polynomial. One interesting observation resulting from the calculations is the fact that a Gauss hypergeometric function recurs in each of sub-integral forms: the first sub-integral form contains zero term of A_n's, the second one contains one term of A_n's, the third one contains two terms of A_n's, etc. In the appendix, I apply the power series expansion and my integral formalism of Heun function to "The 192 solutions of the Heun equation." Due to space restriction final equations for all 192 Heun functions is not included in the paper, but feel free to contact me for the final solutions. Section 5 contains two additional examples using integral forms of Huen function. This paper is 4th out of 10 in series "Special functions and three term recurrence formula (3TRF)". See section 5 for all the papers in the series. The previous paper in series deals with the power series expansion in closed forms of Heun function. The next paper in the series describes analytically the power series expansion of Mathieu function and its integral formalism.

87 pages, Typos corrected in equations in sec.2, 3 images added, 15 pages added. arXiv admin note: substantial text overlap with arXiv:1303.0830, arXiv:1303.0873

References in corpus (16)

• Instability of the massive Klein-Gordon field on the Kerr spacetime
• Heun Functions and Some of Their Applications in Physics
• Heun equation, Teukolsky equation, and type-D metrics
• Examples of Heun and Mathieu functions as solutions of wave equations in curved spaces
• The generalized Heun equation in QFT in curved space-times
• States of the Dirac equation in confining potentials
• Solving systems of transcendental equations involving the Heun functions
• Generalization of the three-term recurrence formula and its applications
• Analytic solution for grand confluent hypergeometric function
• The integral formalism and the generating function of grand confluent hypergeometric function
• The generating functions of Lame equation in Weierstrass's form
• Approximative solution of the spin free Hamiltonian involving only scalar potential for the quark-antiquark system
• Lame equation in the algebraic form
• Power series and integral forms of Lame equation in Weierstrass's form
• Special functions and reversible three-term recurrence formula (R3TRF)
• Complete polynomials using 3-term and reversible 3-term recurrence formulas (3TRF and R3TRF)

Cited by in corpus (9)

• Generalization of the three-term recurrence formula and its applications
• Lame equation in the algebraic form
• Approximative solution of the spin free Hamiltonian involving only scalar potential for the quark-antiquark system
• Analytic solution for grand confluent hypergeometric function
• The integral formalism and the generating function of grand confluent hypergeometric function
• The generating functions of Lame equation in Weierstrass's form
• Power series and integral forms of Lame equation in Weierstrass's form
• Special functions and reversible three-term recurrence formula (R3TRF)
• The violation of a uniqueness theorem and an invariant in the application of Poincaré--Perron theorem to Heun's equation