triconfluent Heun equation

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41: Bibliography W

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Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94 (1), pp. 147–194.

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ISSN 0029-599X, Document, MathReview (Sergey M. Zagorodnyuk), ZentralBlatt

Referenced by:

§2.9(iii).

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Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.

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ISSN 0025-5718, Document, MathReview (B. G. Pachpatte), ZentralBlatt

Referenced by:

§2.9(iii).

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

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Solutions of the fifth Painlevé equation. I. Hokkaido Math. J. 24 (2), pp. 231–267.

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ISSN 0385-4035, FullText, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.10(v).

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On the central connection problem for the double confluent Heun equation. Math. Nachr. 195, pp. 267–276.

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ISSN 0025-584X, MathReview, ZentralBlatt

Referenced by:

§31.12.

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42: Bibliography M

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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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The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.

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ISSN 0025-5718, Document, MathReview (Wadim Zudilin), ZentralBlatt (Masaaki Yoshida)

Referenced by:

§31.3(iii).

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Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.

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ISSN 0305-4470, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.8(vi).

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Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321 (1), pp. 157–195.

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ISSN 0025-5831, Document, MathReview, ZentralBlatt

Referenced by:

§32.9(iii).

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Classical solutions of the third Painlevé equation. Nagoya Math. J. 139, pp. 37–65.

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ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(iii).

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43: Bibliography G

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The Computation of Special Functions by Linear Difference Equations. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 213–243.

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ISBN 90-5699-521-9, MathReview, ZentralBlatt

Referenced by:

§3.6(iii), §3.6(vii).

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Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

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ISBN 0-8176-3837-7, MathReview, ZentralBlatt

Referenced by:

§15.17(v).

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Special classes of solutions of Painlevé equations. Differ. Uravn. 18 (3), pp. 419–429 (Russian).

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In Russian. English translation: Differential Equations 18(3), pp. 317–326.

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ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§32.10(iv), §32.10(v), §32.8(iii), §32.8(v), §32.9(ii).

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Theory of Painlevé’s equations. Differ. Uravn. 11 (11), pp. 373–376 (Russian).

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Notes:

In Russian. English translation: Differential Equations 11(11), pp. 285–287

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.7(iii), §32.7(vi).

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Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions. Phys. Rev. E 47 (4), pp. R2233–R2236.

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Document

Referenced by:

§19.35(ii).

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44: 29.19 Physical Applications

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§29.19(ii) Lamé Polynomials

►Ward (1987) computes finite-gap potentials associated with the periodic Korteweg–de Vries equation. …Hargrave (1978) studies high frequency solutions of the delta wing equation. …Roper (1951) solves the linearized supersonic flow equations. Clarkson (1991) solves nonlinear evolution equations. …

45: Bibliography P

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Sur les équations différentielles du second ordre à points critiques fixès. C.R. Acad. Sc. Paris 143, pp. 1111–1117.

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ZentralBlatt

Referenced by:

§32.2(iv).

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Smoothing of the Stokes phenomenon for high-order differential equations. Proc. Roy. Soc. London Ser. A 436, pp. 165–186.

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ISSN 0962-8444, FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.11(v).

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A uniform asymptotic expansion for the incomplete gamma function. J. Comput. Appl. Math. 148 (2), pp. 323–339.

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ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

(8.11.6), §8.11(iii), (8.12.18), §8.12, §8.12, §8.12, Erratum (V1.0.16) for Equation (8.12.18).

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A new basis for the representation of the rotation group. Lamé and Heun polynomials. J. Mathematical Phys. 14 (8), pp. 1130–1139.

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Document, ZentralBlatt

Referenced by:

§29.18(iv).

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Remarks on computing the probability integral in one and two dimensions. In Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, 1945, 1946, pp. 63–78.

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External Links:

MathReview (L. A. Aroian), ZentralBlatt

Referenced by:

(7.8.8), §7.8, Erratum (V1.0.17) for Equation (7.8.8).

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46: 32.13 Reductions of Partial Differential Equations

§32.13 Reductions of Partial Differential Equations

… ►Equation (32.13.3) also has the similarity reduction … ►

§32.13(ii) Sine-Gordon Equation

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§32.13(iii) Boussinesq Equation

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47: 29.11 Lamé Wave Equation

§29.11 Lamé Wave Equation

►The Lamé (or ellipsoidal) wave equation is given by …In the case $\omega =0$, (29.11.1) reduces to Lamé’s equation (29.2.1). …

48: 9.16 Physical Applications

… ►A quite different application is made in the study of the diffraction of sound pulses by a circular cylinder (Friedlander (1958)). … ►In the study of the stability of a two-dimensional viscous fluid, the flow is governed by the Orr–Sommerfeld equation (a fourth-order differential equation). …An application of Airy functions to the solution of this equation is given in Gramtcheff (1981). ►Airy functions play a prominent role in problems defined by nonlinear wave equations. These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation). …

49: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)

… ►A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3). …

50: 32.12 Asymptotic Approximations for Complex Variables

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§32.12(i) First Painlevé Equation

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§32.12(ii) Second Painlevé Equation

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§32.12(iii) Third Painlevé Equation

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