DLMF: Untitled Document

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Z. Wang and R. Wong (2005) Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.

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External Links:: ISSN 0025-5718, Document, MathReview (B. G. Pachpatte), ZentralBlatt
Referenced by:: §2.9(iii).
Encodings:: BibTeX

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R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:: ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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H. Watanabe (1995) Solutions of the fifth Painlevé equation. I. Hokkaido Math. J. 24 (2), pp. 231–267.

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External Links:: ISSN 0385-4035, FullText, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.10(v).
Encodings:: BibTeX

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A. D. Wheelon (1968) Tables of Summable Series and Integrals Involving Bessel Functions. Holden-Day, San Francisco, CA.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §10.22(vi), §10.23(iv), §10.43(vi).
Encodings:: BibTeX

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H. S. Wilf and D. Zeilberger (1992b) Rational function certification of multisum/integral/“

q

” identities. Bull. Amer. Math. Soc. (N.S.) 27 (1), pp. 148–153.

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External Links:: ISSN 0273-0979, Pdf, Document, MathReview (Robert A. Gustafson), ZentralBlatt
Referenced by:: §16.4(iii).
Encodings:: BibTeX

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W. H. Reid (1995) Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

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External Links:: ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt
Referenced by:: §9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).
Encodings:: BibTeX

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W. H. Reid (1997a) Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

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External Links:: ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt
Referenced by:: (9.11.16), (9.11.17), §9.11(iii), §9.11(v).
Encodings:: BibTeX

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W. H. Reid (1997b) Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.

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External Links:: ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt
Referenced by:: §9.11(iii), §9.11(v).
Encodings:: BibTeX

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S. Ritter (1998) On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.

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External Links:: ISSN 0044-2267, Document, MathReview Entry, ZentralBlatt
Referenced by:: §29.20(ii).
Encodings:: BibTeX

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G. M. Roper (1951) 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).
Encodings:: BibTeX

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A. W. Babister (1967) Transcendental Functions Satisfying Nonhomogeneous Linear Differential Equations. The Macmillan Co., New York.

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External Links:: MathReview (H. Hochstadt), ZentralBlatt
Referenced by:: §11.4(i), §11.4(v), §11.5(ii), §11.5(i), §11.5(ii), §11.5(iii), §11.7(ii), §11.7(iii), §11.7(v), §11.9(i), §11.9(iv), §11.9(iv), Ch.11, §14.29.
Encodings:: BibTeX

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G. E. Barr (1968) A note on integrals involving parabolic cylinder functions. SIAM J. Appl. Math. 16 (1), pp. 71–74.

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External Links:: Document, MathReview (T. Erber), ZentralBlatt
Referenced by:: §12.12.
Encodings:: BibTeX

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A. P. Bassom, P. A. Clarkson, A. C. Hicks, and J. B. McLeod (1992) Integral equations and exact solutions for the fourth Painlevé equation. Proc. Roy. Soc. London Ser. A 437, pp. 1–24.

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External Links:: ISSN 0962-8444, FullText, MathReview (Mehmet Can), ZentralBlatt
Referenced by:: §32.11(v), §32.2(iii).
Encodings:: BibTeX

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S. Bochner (1952) 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).
Encodings:: BibTeX

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M. Brack, M. Mehta, and K. Tanaka (2001) Occurrence of periodic Lamé functions at bifurcations in chaotic Hamiltonian systems. J. Phys. A 34 (40), pp. 8199–8220.

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External Links:: ISSN 0305-4470, Document, MathReview, ZentralBlatt
Referenced by:: §29.19(i).
Encodings:: BibTeX

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A. Dienstfrey and J. Huang (2006) Integral representations for elliptic functions. J. Math. Anal. Appl. 316 (1), pp. 142–160.

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External Links:: ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt
Referenced by:: §23.11.
Encodings:: BibTeX

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B. A. Dubrovin (1981) Theta functions and non-linear equations. Uspekhi Mat. Nauk 36 (2(218)), pp. 11–80 (Russian).

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Notes:: With an appendix by I. M. Krichever. In Russian. English translation: Russian Math. Surveys 36(1981), no. 2, pp. 11–92 (1982)
External Links:: ISSN 0042-1316, Document, MathReview (Alexander A. Pankov), ZentralBlatt
Referenced by:: §21.1, §21.6(ii), §21.7(i), Figure 21.9.2, Figure 21.9.2, §21.9, §21.9, Sidebar 21.SB2, Ch.21, Notations T.
Encodings:: BibTeX

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T. M. Dunster, D. A. Lutz, and R. Schäfke (1993) 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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External Links:: ISSN 0962-8444, FullText, MathReview (W. Balser), ZentralBlatt
Referenced by:: §10.41(iii), §2.8(i).
Encodings:: BibTeX

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T. M. Dunster (1996a) Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function. Proc. Roy. Soc. London Ser. A 452, pp. 1331–1349.

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External Links:: ISSN 0962-8444, FullText, MathReview, ZentralBlatt
Referenced by:: §2.8(vi), §8.12, §8.12.
Encodings:: BibTeX

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T. M. Dunster (2001a) 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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External Links:: ISSN 0022-2526, Document, MathReview, ZentralBlatt
Referenced by:: §10.20(ii), §2.8(i).
Encodings:: BibTeX

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J. K. M. Jansen (1977) Simple-periodic and Non-periodic Lamé Functions. Mathematical Centre Tracts, No. 72, Mathematisch Centrum, Amsterdam.

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External Links:: ISBN 90-6196-130-0, MathReview (V. P. Madan)
Referenced by:: §29.1, §29.1, §29.18(i), §29.19(i), §29.20(i), §29.20(i), §29.22(ii), §29.3(v), §29.6(i), §29.6(ii), §29.6(iii), §29.6(iv), Ch.29, Notations L, Notations L, Notations L, Notations L.
Encodings:: BibTeX

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D. S. Jones, M. J. Plank, and B. D. Sleeman (2010) Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical and Computational Biology Series, CRC Press, Boca Raton, FL.

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Notes:: Second edition of Jones and Sleeman (2003)
External Links:: ISBN 978-1-4200-8357-6, MathReview Entry, ZentralBlatt
Referenced by:: Profile Brian D. Sleeman.
Encodings:: BibTeX

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D. S. Jones and B. D. Sleeman (2003) Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.

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Notes:: A second edition was published in 2010.
External Links:: ISBN 1-58488-296-4, MathReview (Jian Hong Wu), ZentralBlatt
Referenced by:: D. S. Jones, M. J. Plank, and B. D. Sleeman (2010).
Encodings:: BibTeX

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D. S. Jones (1972) Asymptotic behavior of integrals. SIAM Rev. 14 (2), pp. 286–317.

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External Links:: Document, MathReview (A. Erdelyi), ZentralBlatt
Referenced by:: Ch.2.
Encodings:: BibTeX

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N. Joshi and A. V. Kitaev (2001) On Boutroux’s tritronquée solutions of the first Painlevé equation. Stud. Appl. Math. 107 (3), pp. 253–291.

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External Links:: ISSN 0022-2526, Document, MathReview, ZentralBlatt
Referenced by:: §32.12(i).
Encodings:: BibTeX

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N. W. Macfadyen and P. Winternitz (1971) Crossing symmetric expansions of physical scattering amplitudes: The

O(2,1)

group and Lamé functions. J. Mathematical Phys. 12, pp. 281–293.

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External Links:: Document, MathReview
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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J. Meixner (1944) Die Laméschen Wellenfunktionen des Drehellipsoids. Forschungsbericht No. 1952 ZWB (German).

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Notes:: English translation: Lamé wave functions of the ellipsoid of revolution appeared as NACA Technical Memorandum No. 1224, 1949
External Links:: MathReview
Referenced by:: §30.9(iii).
Encodings:: BibTeX

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I. Mező (2020) An integral representation for the Lambert

W

function.

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External Links:: 2012.02480
Referenced by:: (4.13.16), §4.13.
Encodings:: BibTeX

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J. C. P. Miller (1946) The Airy Integral, Giving Tables of Solutions of the Differential Equation

y^{\prime\prime}=xy

. British Association for the Advancement of Science, Mathematical Tables Part-Vol. B, Cambridge University Press, Cambridge.

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External Links:: MathReview (C. J. Bouwkamp), ZentralBlatt
Referenced by:: 1st item, 1st item, §9.2(i), §9.2(ii), §9.2(v), §9.4, (9.6.2), (9.6.3), (9.6.4), (9.6.5), (9.6.6), (9.6.7), (9.6.8), (9.6.9), §9.6(i), §9.8(i), §9.8(ii), §9.8(iv), §9.9(ii), Ch.9.
Encodings:: BibTeX

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L. A. Muraveĭ (1976) Zeros of the function

Ai^{\prime}(z)-\sigma Ai(z)

. Differential Equations 11, pp. 797–811.

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External Links:: ZentralBlatt
Referenced by:: §9.9(i).
Encodings:: BibTeX

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Chapter 1 Additions

The following additions were made in Chapter 1:

Section 1.2

New subsections, 1.2(v) Matrices, Vectors, Scalar Products, and Norms and 1.2(vi) Square Matrices, with Equations (1.2.27)–(1.2.77).
Section 1.3

The title of this section was changed from “Determinants” to “Determinants, Linear Operators, and Spectral Expansions”. An extra paragraph just below (1.3.7). New subsection, 1.3(iv) Matrices as Linear Operators, with Equations (1.3.20), (1.3.21).
Section 1.4

In 1.4(v), three new paragraphs on Stieltjes integrals/measure, with Equations (1.4.23_1)–(1.4.23_3).
Section 1.8

In Subsection 1.8(i), the title of the paragraph “Bessel’s Inequality” was changed to “Parseval’s Formula”. We give the relation between the real and the complex coefficients, and include more general versions of Parseval’s Formula, Equations (1.8.6_1), (1.8.6_2). The title of Subsection 1.8(iv) was changed from “Transformations” to “Poisson’s Summation Formula”, and we added an extra remark just below (1.8.14).
Section 1.10

New subsection, 1.10(xi) Generating Functions, with Equations (1.10.26)–(1.10.29).
Section 1.13

New subsection, 1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms, with Equations (1.13.26)–(1.13.31).
Section 1.14(i)

Another form of Parseval’s formula, (1.14.7_5).
Section 1.16

We include several extra remarks and Equations (1.16.3_5), (1.16.9_5). New subsection, 1.16(ix) References for Section 1.16.
Section 1.17

Two extra paragraphs in Subsection 1.17(ii) Integral Representations, with Equations (1.17.12_1), (1.17.12_2); Subsection 1.17(iv) Mathematical Definitions is almost completely rewritten.
Section 1.18

An entire new section, 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions, including new subsections, 1.18(i)–1.18(x), and several equations, (1.18.1)–(1.18.71).

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Equation (14.6.6)

14.6.6

\mathsf{P}^{-m}_{\nu}\left(x\right)=\left(1-x^{2}\right)^{-m/2}\int_{x}^{1}\!% \dots\!\int_{x}^{1}\mathsf{P}_{\nu}\left(x\right)\left(\!\,\mathrm{d}x\right)^% {m}

The right-hand side has been corrected by replacing the Legendre function $P_{\nu}\left(x\right)$ with the Ferrers function $\mathsf{P}_{\nu}\left(x\right)$ .

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Equations (33.11.2)–(33.11.7)

The arguments of some of the functions in (33.11.2)–(33.11.7) were included to improve clarity of the presentation.

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Equation (15.6.8)

In §15.6, it was noted that (15.6.8) can be rewritten as a fractional integral.

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Equation (14.2.7)

The Wronskian was generalized to include both associated Legendre and Ferrers functions.

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J. Patera and P. Winternitz (1973) 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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External Links:: Document, ZentralBlatt
Referenced by:: §29.18(iv).
Encodings:: BibTeX

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R. Piessens (1982) 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.
External Links:: ISSN 0010-4655, Document, Code
Referenced by:: 6th item.
Encodings:: BibTeX

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R. Piessens and M. Branders (1985) A survey of numerical methods for the computation of Bessel function integrals. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 249–265.

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Notes:: International conference on special functions: theory and computation (Turin, 1984)
External Links:: ISSN 0373-1243, MathReview, ZentralBlatt
Referenced by:: §10.74(vii), §10.74(vii).
Encodings:: BibTeX

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G. Pólya (1949) 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).
Encodings:: BibTeX

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M. Puoskari (1988) A method for computing Bessel function integrals. J. Comput. Phys. 75 (2), pp. 334–344.

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External Links:: ISSN 0021-9991, Document, MathReview (J. G. Herriot), ZentralBlatt
Referenced by:: §10.74(vii), §10.74(vii).
Encodings:: BibTeX

integral equation for Lamé functions

21—28 of 28 matching pages

21: Bibliography W

22: Bibliography R

23: Bibliography B

24: Bibliography D

25: Bibliography J

26: Bibliography M

27: Errata

28: Bibliography P