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1: 10.73 Physical Applications

… ►Bessel functions of the first kind, $J_n\left(x\right)$, arise naturally in applications having cylindrical symmetry in which the physics is described either by Laplace’s equation ${\nabla }^{2}V=0$, or by the Helmholtz equation $({\nabla }^2+k^2)\psi =0$. ►Laplace’s equation governs problems in heat conduction, in the distribution of potential in an electrostatic field, and in hydrodynamics in the irrotational motion of an incompressible fluid. … … ►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). …

2: Bibliography N

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

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§28.26(ii).

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

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An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.

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ISSN 0176-4276, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

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An extension of Laplace’s method. Constr. Approx. 51 (2), pp. 247–272.

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ISSN 0176-4276, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(11.11.16), (11.11.17), §11.11(iii).

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A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

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3: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

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Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

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FullText, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§3.5(viii).

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The Laplace Transform: Theory and Applications. Undergraduate Texts in Mathematics, Springer-Verlag, New York.

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ISBN 0-387-98698-7, MathReview (Philip Heywood), ZentralBlatt

Referenced by:

§1.14(iii), §1.14(vii).

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

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The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.

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ISSN 0253-424X, MathReview

Referenced by:

§9.10(v).

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4: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

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From Klein to Painlevé via Fourier, Laplace and Jimbo. Proc. London Math. Soc. (3) 90 (1), pp. 167–208.

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ISSN 0024-6115, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.9(iii).

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Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§35.5(i).

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Laplace approximations for hypergeometric functions with matrix argument. Ann. Statist. 30 (4), pp. 1155–1177.

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ISSN 0090-5364, Document, MathReview, ZentralBlatt

Referenced by:

§35.6(iv), §35.7(iv), §35.7(iv), §35.9.

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Laplace approximation for Bessel functions of matrix argument. J. Comput. Appl. Math. 155 (2), pp. 359–382.

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ISSN 0377-0427, Document, MathReview (S. Bhargava), ZentralBlatt

Referenced by:

§35.5(ii), §35.5(iii).

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5: Bibliography D

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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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External Links:

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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External Links:

ISSN 0037-959X, MathReview, ZentralBlatt

Referenced by:

§31.12.

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Handbuch der Laplace-Transformation. Bd. II. Anwendungen der Laplace-Transformation. 1. Abteilung. Birkhäuser Verlag, Basel und Stuttgart (German).

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External Links:

MathReview (I. I. Hirschman), ZentralBlatt

Referenced by:

§2.5(i).

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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2nd item.

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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

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6: Bibliography K

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:

ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

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1st item.

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

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The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

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A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Mir, Moscow.

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Notes:

Translated from the Russian by George Yankovsky. Reprint of the 1977 translation

External Links:

MathReview, ZentralBlatt

Referenced by:

§3.5(viii).

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7: Bibliography O

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Tables of Laplace Transforms. Springer-Verlag, Berlin-New York.

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Notes:

Table errata: Math. Comp. v. 50 (1988), no. 182, pp. 653–654.

External Links:

ISBN 0-387-06350-1, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.43(vi), §11.5(iii), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(ii), §13.23(i), §14.17(v), §15.14, §18.17(ix), §20.10(ii), §5.13, §6.14(iii), §6.7(iv), §7.14(iii), §7.7(iii), §8.6(iii).

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Studies on the Painlevé equations. I. Sixth Painlevé equation $P_{\mathrm{VI}}$ . Ann. Mat. Pura Appl. (4) 146, pp. 337–381.

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External Links:

ISSN 0003-4622, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(vi), §32.6(v), §32.7(vii), §32.7(viii).

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Studies on the Painlevé equations. II. Fifth Painlevé equation $P_{\mathrm{V}}$ . Japan. J. Math. (N.S.) 13 (1), pp. 47–76.

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External Links:

ISSN 0289-2316, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(v), §32.6(v), §32.7(viii).

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Studies on the Painlevé equations. IV. Third Painlevé equation $P_{\mathrm{III}}$ . Funkcial. Ekvac. 30 (2-3), pp. 305–332.

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External Links:

ISSN 0532-8721, FullText, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.10(iii), §32.2(iii), §32.6(iv), §32.7(viii).

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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

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8: Bibliography G

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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

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Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.

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Notes:

Problem proposed by P. Smith.

External Links:

Document

Referenced by:

(9.10.14), (9.10.15), (9.10.16), §9.10(v).

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

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ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

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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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External Links:

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

Referenced by:

§15.17(v).

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9: 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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External Links:

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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External Links:

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

Referenced by:

§2.11(v).

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Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:

Double-precision Fortran, maximum accuracy 20S.

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ISSN 0010-4655, Document, Code

Referenced by:

6th item.

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Integrals and Series: Direct Laplace Transforms, Vol. 4. Gordon and Breach Science Publishers, New York.

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Notes:

Table erratum: Math. Comp. v. 66 (1997), no. 220, p. 1766.

External Links:

ISBN 2-88124-837-3, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§1.14(viii), §1.4(iv), §10.22(vi), §10.43(vi), §10.71(iii), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(ii), §13.10(vi), §13.23(i), §14.17(v), §15.14, §16.20, §16.5, §18.17(ix), §19.13(iii), §20.10(iii), §24.13(iii), §25.11(ix), §25.12(ii), §25.7, §5.13, §6.14(iii), §7.14(iii), §8.14, §9.10(ix).

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Integrals and Series: Inverse Laplace Transforms, Vol. 5. Gordon and Breach Science Publishers, New York.

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External Links:

ISBN 2-88124-838-1, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§1.14(viii), §1.4(iv), §10.22(vi), §10.43(vi), §10.71(iii), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(ii), §13.10(vi), §13.23(i), §13.23(iv), §14.17(v), §15.14, §15.14, §16.15, §16.20, §16.5, §18.17(ix), §19.13(iii), §24.13(iii), §25.11(ix), §25.12(ii), §25.7, §5.13, §6.14(iii), §7.14(iii), §8.6(iii), §9.10(ix).

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