Heun functions and Heun equation

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1: 31.1 Special Notation

… ►Sometimes the parameters are suppressed.

2: 31.6 Path-Multiplicative Solutions

… ►A further extension of the notation (31.4.1) and (31.4.3) is given by …

3: 31.4 Solutions Analytic at Two Singularities: Heun Functions

§31.4 Solutions Analytic at Two Singularities: Heun Functions

… ►To emphasize this property this set of functions is denoted by … ►The eigenvalues $q_m$ satisfy the continued-fraction equation … ►The set $q_m$ depends on the choice of $s_1$ and $s_2$. ►The solutions (31.4.3) are called the Heun functions. …

4: 31.12 Confluent Forms of Heun’s Equation

… ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. ►Mathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions (§30.12) are special cases of solutions of the confluent Heun equation. …

5: 31.13 Asymptotic Approximations

§31.13 Asymptotic Approximations

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6: 31.17 Physical Applications

§31.17 Physical Applications

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§31.17(ii) Other Applications

►Heun functions appear in the theory of black holes (Kerr (1963), Teukolsky (1972), Chandrasekhar (1984), Suzuki et al. (1998), Kalnins et al. (2000)), lattice systems in statistical mechanics (Joyce (1973, 1994)), dislocation theory (Lay and Slavyanov (1999)), and solution of the Schrödinger equation of quantum mechanics (Bay et al. (1997), Tolstikhin and Matsuzawa (2001), and Hall et al. (2010)). ►For applications of Heun’s equation and functions in astrophysics see Debosscher (1998) where different spectral problems for Heun’s equation are also considered. …

7: Gerhard Wolf

… ►Wolf has published papers on Mathieu functions, orthogonal polynomials, and Heun functions. His book Mathieu Functions and Spheroidal Functions and Their Mathematical Foundations: Further Studies (with J. … Schmidt) of the Chapter Double Confluent Heun Equation in the book Heun’s Differential Equations (A. … ►

28 Mathieu Functions and Hill’s Equation

►Wolf served as a Validator for the original release and publication in May 2010 of the NIST Digital Library of Mathematical Functions and the NIST Handbook of Mathematical Functions.

8: 31.18 Methods of Computation

§31.18 Methods of Computation

… ►The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 28–30.

9: 31.9 Orthogonality

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§31.9(i) Single Orthogonality

… ►For corresponding orthogonality relations for Heun functions (§31.4) and Heun polynomials (§31.5), see Lambe and Ward (1934), Erdélyi (1944), Sleeman (1966a), and Ronveaux (1995, Part A, pp. 59–64). ►

§31.9(ii) Double Orthogonality

►Heun polynomials $w_j={\mathrm{𝐻𝑝}}_{n_j,m_j}$, $j=1,2$, satisfy …and the integration paths ${ℒ}_1$, ${ℒ}_2$ are Pochhammer double-loop contours encircling distinct pairs of singularities $\{0,1\}$, $\{0,a\}$, $\{1,a\}$. …

10: 31.10 Integral Equations and Representations

§31.10 Integral Equations and Representations

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Kernel Functions

… ►Then the integral equation (31.10.1) is satisfied by $w(z)=w_m(z)$ and $W(z)={\kappa }_m{w}_m(z)$, where $w_m(z)=(0,1){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$ and ${\kappa }_m$ is the corresponding eigenvalue. … ►

Kernel Functions

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