Liouville transformation for differential equations

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11: 2.6 Distributional Methods

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§2.6(ii) Stieltjes Transform

… ►The Stieltjes transform of $f(t)$ is defined by … ►Corresponding results for the generalized Stieltjes transform … ►The Riemann–Liouville fractional integral of order $\mu$ is defined by … ►For rigorous derivations of these results and also order estimates for ${\delta }_n(x)$, see Wong (1979) and Wong (1989, Chapter 6).

12: Bibliography D

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Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. CWI Monographs, Vol. 2, North-Holland Publishing Co., Amsterdam.

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ISBN 0-444-87634-0, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§3.7(v).

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Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions. Proc. Roy. Soc. London Ser. A 440, pp. 37–54.

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ISSN 0962-8444, FullText, MathReview (W. Balser), ZentralBlatt

Referenced by:

§10.41(iii), §2.8(i).

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Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one. Methods Appl. Anal. 3 (1), pp. 109–134.

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ISSN 1073-2772, Pdf, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§2.11(v).

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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.

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ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§10.20(ii), §2.8(i).

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Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.

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ISSN 0219-5305, Document, Link, MathReview (Thomas Mbah Acho), ZentralBlatt

Referenced by:

5th item, §14.32.

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13: Bibliography L

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The two-point connection problem for differential equations of the Heun class. Teoret. Mat. Fiz. 101 (3), pp. 360–368 (Russian).

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Notes:

In Russian. English translation: Theoret. and Math. Phys. 101(1994), no. 3, pp. 1413–1418 (1995)

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ISSN 0564-6162, Document, MathReview, ZentralBlatt

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§31.18.

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Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.

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Document, ZentralBlatt

Referenced by:

§31.10(i), §31.9(i).

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Algorithm 537: Characteristic values of Mathieu’s differential equation. ACM Trans. Math. Software 5 (1), pp. 112–117.

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ISSN 0098-3500, Document, GAMS (module 537; package TOMS)

Referenced by:

2nd item, 2nd item.

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Some Differential Equations Satisfied by Hypergeometric Functions. In Approximation and Computation (West Lafayette, IN, 1993), Internat. Ser. Numer. Math., Vol. 119, pp. 371–381.

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MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§16.8(ii).

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The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).

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Notes:

In Russian. English translation: Differential Equations 7(6), pp. 853–854

External Links:

MathReview (Z. Rubinstein), ZentralBlatt

Referenced by:

§32.7(ii).

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14: Bibliography G

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The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals. Ann. of Math. (2) 48 (4), pp. 785–826.

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FullText, MathReview (E. T. Copson), ZentralBlatt

Referenced by:

§35.2, §35.3(ii).

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Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments. ACM Trans. Math. Software 30 (2), pp. 145–158.

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ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(viii).

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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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Notes:

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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