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11: 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). Such a solution is given in terms of a Riemann theta function with two phases. …The agreement of these solutions with two-dimensional surface water waves in shallow water was considered in Hammack et al. (1989, 1995).

12: Bibliography O

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Exponentially improved asymptotic solutions of ordinary differential equations. II Irregular singularities of rank one. Proc. Roy. Soc. London Ser. A 445, pp. 39–56.

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External Links:

ISSN 0962-8444, FullText, MathReview (Hussain S. Nur), ZentralBlatt

Referenced by:

§13.4(i), §2.11(v), §2.7(ii).

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Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.

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ISSN 1364-5021, Document, MathReview (Donald A. Lutz), ZentralBlatt

Referenced by:

§2.11(v), §2.7(ii).

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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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Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

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Error bounds for asymptotic solutions of second-order differential equations having an irregular singularity of arbitrary rank. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 244–249.

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External Links:

Document, MathReview, ZentralBlatt

Referenced by:

§2.7(ii).

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Asymptotic solutions of linear ordinary differential equations at an irregular singularity of rank unity. Methods Appl. Anal. 4 (4), pp. 375–403.

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External Links:

ISSN 1073-2772, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§2.7(ii).

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13: 18.39 Applications in the Physical Sciences

… ►Also presented are the analytic solutions for the $L^2$, bound state, eigenfunctions and eigenvalues of the Morse oscillator which also has analytically known non-normalizable continuum eigenfunctions, thus providing an example of a mixed spectrum. … ►The solutions of (18.39.8) are subject to boundary conditions at $a$ and $b$. … ►The solutions (18.39.8) are called the stationary states as separation of variables in (18.39.9) yields solutions of form … ►Brief mention of non-unit normalized solutions in the case of mixed spectra appear, but as these solutions are not OP’s details appear elsewhere, as referenced. … ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. …

14: Bibliography R

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On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

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ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§16.4(iii), §16.4(iii).

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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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External Links:

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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External Links:

MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

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Some Applications of the Lamé Function Solutions of the Linearised Supersonic Flow Equations. Technical Reports and Memoranda Technical Report 2865, Aeronautical Research Council (Great Britain).

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Notes:

Reprinted by Her Majesty’s Stationery Office, London, 1962.

External Links:

MathReview (P. Germain)

Referenced by:

§29.19(ii).

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15: 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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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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Exact analytical solutions of $z{e}^z=a$ . J. Math. Anal. Appl. 43 (3), pp. 626–632.

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Document, MathReview (Pl. Kannappan), ZentralBlatt

Referenced by:

§4.13.

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The linear differential equation whose solutions are the products of solutions of two given differential equations. J. Math. Anal. Appl. 98 (1), pp. 130–147.

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ISSN 0022-247X, Document, FullText, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§1.13(v).

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Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

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Notes:

Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.

External Links:

ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt

Referenced by:

§3.3(vi), §3.4(i), §3.5(i).

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16: Bibliography P

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Computer algebra system designed for computations in number theory. Many transcendental functions are also included. PARI is also available as a C library to allow for faster computations. Originally developed by Henri Cohen and co-workers at the Université Bordeaux I.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.

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Tables of the Incomplete Beta-function. 2nd edition, Published for the Biometrika Trustees at the Cambridge University Press, Cambridge.

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External Links:

MathReview, ZentralBlatt

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1st item.

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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

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6th item.

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Fast analytic formulas for the modified Bessel functions of imaginary order for spectral line broadening calculations. J. Quantit. Spec. and Rad. Trans. 62 (4), pp. 389–395.

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External Links:

Document

Referenced by:

§10.76(ii).

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Numerical Solution of Sturm-Liouville Problems. Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York.

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External Links:

ISBN 0-19-853415-9, MathReview (L. Greenberg), ZentralBlatt

Referenced by:

§3.7(iv).

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17: 1.13 Differential Equations

… ►The equation … ►Two solutions $w_1(z)$ and $w_2(z)$ are called a fundamental pair if any other solution $w(z)$ is expressible as … ►Assume that in the equation … ►The inhomogeneous (or nonhomogeneous) equation … ►The product of any two solutions of (1.13.1) satisfies …

18: 31.13 Asymptotic Approximations

… ►For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). ►For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).

19: Tom M. Apostol

… ► 2016) was Professor Emeritus in the Department of Mathematics at the California Institute of Technology (Caltech), Pasadena, California, until his death on May 8, 2016. … ►Apostol was born on August 20, 1923. …In 1950, he arrived at Caltech as an assistant professor; he was named associate professor in 1956, professor in 1962, and professor emeritus in 1992. … ►He was internationally known for his textbooks on calculus, analysis, and analytic number theory, which have been translated into five languages, and for creating Project MATHEMATICS!, a series of video programs that bring mathematics to life with computer animation, live action, music, and special effects. The videos captured first-place honors at a dozen international video festivals, and have been translated into Hebrew, Portuguese, French, and Spanish. …

20: 31.1 Special Notation

… ►Sometimes the parameters are suppressed.