relations to Heun functions

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1: 31.7 Relations to Other Functions

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§31.7(i) Reductions to the Gauss Hypergeometric Function

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§31.7(ii) Relations to Lamé Functions

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2: 31.8 Solutions via Quadratures

… ►For $𝐦=(m_0,0,0,0)$, these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. …

3: 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 $z=\mathrm{\infty }$. …

4: 29.2 Differential Equations

… ►For the Weierstrass function $\mathrm{\wp }$ see §23.2(ii). …

5: Bibliography V

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An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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External Links:

ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt

Referenced by:

§31.10(i).

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Airy Functions and Applications to Physics. Second edition, Imperial College Press, London.

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ISBN 978-1-84816-548-9; 1-84816-548-X, MathReview Entry, ZentralBlatt

Referenced by:

§1.17(ii), §9.10(ii), §9.10(ix), §9.11(v), §9.16, §9.16, §9.5(i).

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Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions. CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam.

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Notes:

A reprint was published in 1986.

External Links:

ISBN 90-6196-277-3, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

§5.21, §6.18(iv), §7.22(v), §7.6(i), §7.7(i), §7.7(ii).

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Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

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ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8, §29.8.

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Asymptotic expansion of the generalized hypergeometric function ${}_p{}^{}F_q^{}(z)$ as $z\to \mathrm{\infty }$ for . Anal. Appl. (Singap.) 21 (2), pp. 535–545.

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ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§16.11(i), §16.11, Erratum (V1.1.9) for Section 16.11(i).

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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: 31.3 Basic Solutions

… ► $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ denotes the solution of (31.2.1) that corresponds to the exponent $0$ at $z=0$ and assumes the value $1$ there. … ►

§31.3(ii) Fuchs–Frobenius Solutions at Other Singularities

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§31.3(iii) Equivalent Expressions

… ►Each is related to the solution (31.3.1) by one of the automorphisms of §31.2(v). …For example, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ is equal to …

8: Bibliography M

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Chebyshev expansions for modified Struve and related functions. Math. Comp. 60 (202), pp. 735–747.

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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2nd item.

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On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.

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ISSN 0022-0396, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§31.7(i).

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Uniform computation of the error function and other related functions. Math. Comp. 25 (114), pp. 339–344.

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FullText, MathReview, ZentralBlatt

Referenced by:

§7.22(i).

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Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.

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ISSN 0036-1429, Document, MathReview (M. Brannigan), ZentralBlatt

Referenced by:

§4.45(ii).

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A new symmetry related to $\mathrm{𝑆𝑈}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

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ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.15.

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9: Bibliography S

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Chebyshev expansions for the error and related functions. Math. Comp. 32 (144), pp. 1232–1240.

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FullText, Document, MathReview (L. Fox), ZentralBlatt

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3rd item.

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Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros. Math. Comp. 70 (235), pp. 1205–1220.

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ISSN 0025-5718, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.74(vi).

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Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§31.13.

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Integral equations and relations for Lamé functions and ellipsoidal wave functions. Proc. Cambridge Philos. Soc. 64, pp. 113–126.

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Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§29.8.

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Non-linear integral equations for Heun functions. Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.

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Document, MathReview (J. D. DePree), ZentralBlatt

Referenced by:

§31.10(ii).

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10: Bibliography T

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Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. SIAM J. Math. Anal. 21 (1), pp. 241–261.

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ISSN 0036-1410, Document, MathReview (Pablo Martín), ZentralBlatt

Referenced by:

§13.8(iii), §13.8(iii).

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Special Functions: An Introduction to the Classical Functions of Mathematical Physics. John Wiley & Sons Inc., New York.

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ISBN 0-471-11313-1, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§1.13(iv), §10.23(iii), Ch.12, §13.14(ii), §13.19, §13.2(i), §13.2(iii), §13.2(vii), §13.6(i), §13.6(ii), §13.6(iii), §13.6(iv), §13.7(i), Ch.13, §14.31(iii), Ch.14, §15.2(ii), §15.4(i), Ch.15, §24.17(i), §24.9, §3.5(i), §5.11(iii), §5.12, §5.19(i), §5.2(i), §5.2(ii), §5.5(i), §5.5(iii), §5.7(ii), (5.9.2_5), §5.9(i), §5.9(ii), Ch.5, §6.11, §6.16(i), §6.7(iii), Ch.6, §7.2(i), §7.2(ii), §7.2(iii), §7.20(ii), Ch.7, §8.11(i), §8.17(ii), §8.17(iii), §8.17(iv), §8.17(v), §8.18(ii), §8.18(ii), §8.18(ii), §8.19(i), §8.25(iv), §8.6(i), §8.6(iii), Ch.8, Ch.9, Profile Nico M. Temme.

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Modified Bessel functions $I_{\nu }(z)$ and $K_{\nu }(z)$ of real order and complex argument, to selected accuracy. Comput. Phys. Comm. 47 (2-3), pp. 245–257.

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Notes:

For erratum see same journal 159 (2004), no. 3, p. 243. Double-precision Fortran, accuracy 24S, but can be increased.

External Links:

ISSN 0010-4655, Document, Code, MathReview (John P. Coleman)

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3rd item, 4th item.

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Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

§33.23(vi), I. J. Thompson and A. R. Barnett (1985).

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Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.

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External Links:

Document

Referenced by:

§31.17(ii).

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