triconfluent Heun equation

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31: Bibliography F

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The Lamé wave equation. Uspekhi Mat. Nauk 44 (1(265)), pp. 123–144, 248 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 44(1989), no. 1, pp. 153–180

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ISSN 0042-1316, Document, MathReview (R. G. Langebartel), ZentralBlatt

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§29.11.

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Asymptotics of the spectrum of the Heun equation and of Heun functions. Izv. Akad. Nauk SSSR Ser. Mat. 55 (3), pp. 631–646 (Russian).

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Notes:

In Russian. English translation: Math. USSR-Izv., 38(1992), no. 3, pp. 621–635 (1992)

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ISSN 0373-2436, Document, MathReview (Andreĭ Bolibrukh), ZentralBlatt

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§31.13.

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On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

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ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.13(i), §32.7(i), §32.7(iii).

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From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.

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ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§32.7(ii).

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Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.

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MathReview (W. J. Trjitzinsky), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations S, Notations S.

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32: 28.18 Integrals and Integral Equations

§28.18 Integrals and Integral Equations

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33: Errata

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Equation (16.16.5_5).

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31.3.10 $$z^{-\alpha }H\mathrm{\ell }(\frac{1}{a},\frac{q}{a}-\alpha \left(\beta -ϵ\right)-\frac{\alpha }{a}\left(\beta -\delta \right);\alpha ,\alpha -\gamma +1,\alpha -\beta +1,\delta ;\frac{1}{z})$$

31.3.11 $$z^{-\beta }H\mathrm{\ell }(\frac{1}{a},\frac{q}{a}-\beta \left(\alpha -ϵ\right)-\frac{\beta }{a}\left(\alpha -\delta \right);\beta ,\beta -\gamma +1,\beta -\alpha +1,\delta ;\frac{1}{z})$$

In both equations, the second entry in the $H\mathrm{\ell }$ has been corrected with an extra minus sign.

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31.16.2

$xy=a{\sin }^2\theta {\cos }^2\varphi ,$

$(x-1)(y-1)=(1-a){\sin }^2\theta {\sin }^2\varphi ,$

$(x-a)(y-a)=a(a-1){\cos }^2\theta$

31.16.3 $$A_0=\frac{n!}{{\left(\gamma +\delta \right)}_n}{\mathrm{𝐻𝑝}}_{n,m}\left(1\right),Q_0{A}_0+R_0{A}_1=0$$

Originally $x,y$ were incorrectly defined by the set of equations (31.16.2), given previously as “$x={\sin }^2\theta {\cos }^2\varphi ,$ $y={\sin }^2\theta {\sin }^2\varphi$”. In fact, $x,y$ are implicitly defined by the corrected set of equations. In (31.16.3), the initial data $A_0$, previously missing, has now been included.

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Four of the terms were rewritten for improved clarity.

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has been generalized to cover an additional case.

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

Referenced by:

§31.10(i).

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A Mathieu equation for ships rolling among waves. I, II. Norske Vid. Selsk. Forh., Trondheim 22 (25–26), pp. 113–123.

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MathReview (J. V. Wehausen), ZentralBlatt

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2nd item.

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On the growth of convergence radii for the eigenvalues of the Mathieu equation. Math. Nachr. 192, pp. 239–253.

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ISSN 0025-584X, Document, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§28.6(i).

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

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On the rational solutions of the second Painlevé equation. Differ. Uravn. 1 (1), pp. 79–81 (Russian).

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Notes:

In Russian. English translation: Differential Equations 1(1), pp. 58–59.

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MathReview, ZentralBlatt

Referenced by:

§32.8(ii).

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35: Bibliography T

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Rotating black holes: Separable wave equations for gravitational and electromagnetic perturbations. Phys. Rev. Lett. 29 (16), pp. 1114–1118.

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Document

Referenced by:

§31.17(ii).

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Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.

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MathReview (H. Pollard)

Referenced by:

§1.18(x).

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Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.

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ISBN 0198533187, MathReview Entry

Referenced by:

§1.18(x).

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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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Document

Referenced by:

§31.17(ii).

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Iterative Methods for the Solution of Equations. Prentice-Hall Series in Automatic Computation, Prentice-Hall Inc., Englewood Cliffs, N.J..

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MathReview (W. C. Rheinboldt), ZentralBlatt

Referenced by:

§3.8(iii).

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36: Bibliography D

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Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes. Phys. Rev. E (3) 57 (1), pp. 252–275.

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ISSN 1063-651X, Document, MathReview (Victor I. Stepanov), ZentralBlatt

Referenced by:

§31.17(ii).

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Formes canoniques des équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (1-2), pp. 53–78.

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MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§31.12.

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Sur les équations confluentes de l’équation de Heun. Ann. Soc. Sci. Bruxelles Sér. I 92 (3), pp. 151–189.

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ISSN 0037-959X, MathReview, ZentralBlatt

Referenced by:

§31.12.

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The KP equation with quasiperiodic initial data. Phys. D 123 (1-4), pp. 123–152.

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Notes:

Nonlinear waves and solitons in physical systems (Los Alamos, NM, 1997)

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ISSN 0167-2789, Document, MathReview (Pavel Krejčí), ZentralBlatt

Referenced by:

§21.9.

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Pole dynamics for elliptic solutions of the Korteweg-de Vries equation. Math. Phys. Anal. Geom. 3 (1), pp. 49–74.

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ISSN 1385-0172, Document, MathReview (Giulio Soliani), ZentralBlatt

Referenced by:

§23.21(ii).

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37: Vadim B. Kuznetsov

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31 Heun Functions

38: Bibliography B

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An Introduction to Linear Difference Equations. Dover Publications Inc., New York.

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Notes:

Unaltered republication of the original 1927 Harvard University edition.

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MathReview, ZentralBlatt

Referenced by:

§8.1, Notations P, Notations Q.

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Normalization integrals of orthogonal Heun functions. J. Math. Phys. 38 (7), pp. 3692–3699.

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ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§31.9(i).

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Vortices in Ginzburg-Landau Equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 11–19.

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ISSN 1431-0643, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§17.17.

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Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation. J. Phys. A 30 (2), pp. 559–571.

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ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§31.13.

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The double confluent Heun equation: Characteristic exponent and connection formulae. Methods Appl. Anal. 1 (3), pp. 348–370.

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ISSN 1073-2772, FullText, MathReview (W. Balser), ZentralBlatt

Referenced by:

§31.12.

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39: Bibliography H

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Solving Ordinary Differential Equations. I. Nonstiff Problems. 2nd edition, Springer Series in Computational Mathematics, Vol. 8, Springer-Verlag, Berlin.

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Notes:

A reprinting with corrections appeared in 2000.

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ISBN 3-540-56670-8, MathReview, ZentralBlatt

Referenced by:

§3.7(v), §3.7(i).

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Soft-core Coulomb potentials and Heun’s differential equation. J. Math. Phys. 51 (2), pp. Art. ID 022107, 19 pages.

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ISSN 0022-2488, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§31.17(ii), §31.17(ii).

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Poncelet Polygons and the Painlevé Equations. In Geometry and Analysis (Bombay, 1992), Ramanan (Ed.), pp. 151–185.

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MathReview (Henrik Pedersen), ZentralBlatt

Referenced by:

§32.9(iii).

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Estimates of the stability intervals for Hill’s equation. Proc. Amer. Math. Soc. 14 (6), pp. 930–932.

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FullText, MathReview (P. Ungar), ZentralBlatt

Referenced by:

§28.29(iii).

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Differential Equations: A Modern Approach. Holt, Rinehart and Winston, New York.

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Notes:

Reprinted by Dover in 1975.

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MathReview (Z. Opial), ZentralBlatt

Referenced by:

§29.3(vi).

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40: Bibliography R

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Composite approximations to the solutions of the Orr-Sommerfeld equation. Studies in Appl. Math. 51, pp. 341–368.

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MathReview (E. A. Boyd), ZentralBlatt

Referenced by:

(9.13.25), (9.13.26), (9.13.28), (9.13.29), (9.13.32), (9.13.33), (9.13.34), §9.13(ii), §9.13(ii).

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Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.

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MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

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Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.

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MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

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Asymptotics and Bounds of the Roots of Equations (Russian). Zinatne, Riga.

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External Links:

ISBN 5-7966-0570-4, MathReview (Guillermo López Lagomasino), ZentralBlatt

Referenced by:

§12.11(ii), §6.13.

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Heun’s Differential Equations. The Clarendon Press Oxford University Press, New York.

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ISBN 0-19-859695-2, MathReview (Harold Exton), ZentralBlatt

Referenced by:

§31.1, §31.12, §31.13, §31.2(v), §31.2(i), §31.2(iv), §31.3(ii), §31.4, §31.7(ii), §31.9(i), Ch.31, Profile Gerhard Wolf.

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