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1: Bibliography S
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  • A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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    External Links:
    Document, MathReview (F. W. J. Olver), ZentralBlatt
    Referenced by:
    §28.8(iv).
    Encodings:
    BibTeX
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  • C. Snow (1952) Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, D.C..
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    External Links:
    MathReview, ZentralBlatt
    Referenced by:
    §13.23(iv), Ch.14, §31.3(iii).
    Encodings:
    BibTeX
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  • I. A. Stegun and R. Zucker (1970) Automatic computing methods for special functions. I. J. Res. Nat. Bur. Standards Sect. B 74B, pp. 211–224.
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    Notes:
    Includes Fortran code, maximum accuracy 18S.
    External Links:
    MathReview (Bernard H. Rosman), ZentralBlatt
    Referenced by:
    §7.25(ii).
    Encodings:
    BibTeX
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  • F. Stenger (1966a) Error bounds for asymptotic solutions of differential equations. I. The distinct eigenvalue case. J. Res. Nat. Bur. Standards Sect. B 70B, pp. 167–186.
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    External Links:
    MathReview (F. W. J. Olver), ZentralBlatt
    Referenced by:
    §2.7(ii).
    Encodings:
    BibTeX
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  • F. Stenger (1966b) Error bounds for asymptotic solutions of differential equations. II. The general case. J. Res. Nat. Bur. Standards Sect. B 70B, pp. 187–210.
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    External Links:
    MathReview (F. W. J. Olver), ZentralBlatt
    Referenced by:
    §2.7(ii).
    Encodings:
    BibTeX
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    2: Bibliography N
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  • National Bureau of Standards (1944) Tables of Lagrangian Interpolation Coefficients. Columbia University Press, New York.
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    Notes:
    Technical Director: Arnold N. Lowan
    External Links:
    MathReview (W. Feller), ZentralBlatt
    Referenced by:
    §3.3(ii).
    Encodings:
    BibTeX
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  • National Bureau of Standards (1958) Integrals of Airy Functions. National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..
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    External Links:
    MathReview (J. C. P. Miller), ZentralBlatt
    Referenced by:
    2nd item, 3rd item.
    Encodings:
    BibTeX
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  • National Bureau of Standards (1967) Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..
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    External Links:
    MathReview, ZentralBlatt
    Referenced by:
    §28.1, §28.26(i), 6th item, Notations S, Notations S.
    Encodings:
    BibTeX
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  • M. Newman (1967) Solving equations exactly. J. Res. Nat. Bur. Standards Sect. B 71B, pp. 171–179.
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    External Links:
    MathReview, ZentralBlatt
    Referenced by:
    §27.15.
    Encodings:
    BibTeX
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  • E. W. Ng and M. Geller (1969) 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).
    Encodings:
    BibTeX
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    3: Bibliography K
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  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) 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
    Referenced by:
    1st item.
    Encodings:
    BibTeX
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  • M. K. Kerimov (1980) 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).
    Encodings:
    BibTeX
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  • A. V. Kitaev and A. H. Vartanian (2004) 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).
    Encodings:
    BibTeX
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  • T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.
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    External Links:
    ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt
    Referenced by:
    §18.26(v), §18.26(v).
    Encodings:
    BibTeX
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  • Y. A. Kravtsov (1964) Asymptotic solution of Maxwell’s equations near caustics. Izv. Vuz. Radiofiz. 7, pp. 1049–1056.
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    Referenced by:
    §36.14(i).
    Encodings:
    BibTeX
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    4: Bibliography
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  • M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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    External Links:
    Document, MathReview (Mark Adler)
    Referenced by:
    §32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.
    Encodings:
    BibTeX
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  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
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    External Links:
    ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt
    Referenced by:
    §24.12(iii).
    Encodings:
    BibTeX
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  • H. Airault (1979) Rational solutions of Painlevé equations. Stud. Appl. Math. 61 (1), pp. 31–53.
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    External Links:
    ISSN 0022-2526, MathReview (H. Hochstadt), ZentralBlatt
    Referenced by:
    §32.10(ii), §32.8(i), §32.8(v).
    Encodings:
    BibTeX
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  • D. E. Amos (1989) Repeated integrals and derivatives of K Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.
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    External Links:
    ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt
    Referenced by:
    §10.43(iii).
    Encodings:
    BibTeX
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  • F. M. Arscott (1956) Perturbation solutions of the ellipsoidal wave equation. Quart. J. Math. Oxford Ser. (2) 7, pp. 161–174.
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    External Links:
    ISSN 0033-5606, Document, MathReview (A. Erdélyi), ZentralBlatt
    Referenced by:
    §29.11.
    Encodings:
    BibTeX
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    5: Bibliography O
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  • S. Okui (1974) Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.
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    External Links:
    ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt
    Referenced by:
    §10.22(vi), §10.43(vi).
    Encodings:
    BibTeX
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  • S. Okui (1975) Complete elliptic integrals resulting from infinite integrals of Bessel functions. II. J. Res. Nat. Bur. Standards Sect. B 79B (3-4), pp. 137–170.
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    External Links:
    ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt
    Referenced by:
    §10.22(vi), §10.43(vi).
    Encodings:
    BibTeX
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  • F. W. J. Olver (1967a) Numerical solution of second-order linear difference equations. J. Res. Nat. Bur. Standards Sect. B 71B, pp. 111–129.
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    External Links:
    MathReview (G. N. Lance), ZentralBlatt
    Referenced by:
    §3.6(iv), §3.6(v).
    Encodings:
    BibTeX
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  • F. W. J. Olver (1967b) Bounds for the solutions of second-order linear difference equations. J. Res. Nat. Bur. Standards Sect. B 71B (4), pp. 161–166.
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    External Links:
    MathReview (C. W. Clenshaw), ZentralBlatt
    Referenced by:
    §2.9(i).
    Encodings:
    BibTeX
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  • F. W. J. Olver (1976) Improved error bounds for second-order differential equations with two turning points. J. Res. Nat. Bur. Standards Sect. B 80B (4), pp. 437–440.
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    External Links:
    MathReview (Peter Deuflhard), ZentralBlatt
    Referenced by:
    §2.8(vi).
    Encodings:
    BibTeX
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    6: Bibliography G
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  • M. Geller and E. W. Ng (1969) A table of integrals of the exponential integral. J. Res. Nat. Bur. Standards Sect. B 73B, pp. 191–210.
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    External Links:
    MathReview (John Todd), ZentralBlatt
    Referenced by:
    §6.14(iii), §6.17, §6.7(iv).
    Encodings:
    BibTeX
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  • M. Geller and E. W. Ng (1971) A table of integrals of the error function. II. Additions and corrections. J. Res. Nat. Bur. Standards Sect. B 75B, pp. 149–163.
    β“˜
    Notes:
    See for the first part same journal, Sect. B 73, 1-20 (1969)
    External Links:
    MathReview, ZentralBlatt
    Referenced by:
    §7.14(iii).
    Encodings:
    BibTeX
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  • A. Gil, J. Segura, and N. M. Temme (2014) 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
    External Links:
    ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
    Referenced by:
    3rd item, §10.77(ix).
    Encodings:
    BibTeX
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  • M. L. Glasser (1976) Definite integrals of the complete elliptic integral K . J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 313–323.
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    External Links:
    MathReview (H. E. Fettis), ZentralBlatt
    Referenced by:
    §19.13(i).
    Encodings:
    BibTeX
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  • K. Goldberg, F. T. Leighton, M. Newman, and S. L. Zuckerman (1976) Tables of binomial coefficients and Stirling numbers. J. Res. Nat. Bur. Standards Sect. B 80B (1), pp. 99–171.
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    External Links:
    MathReview (M. S. Cheema), ZentralBlatt
    Referenced by:
    §26.21.
    Encodings:
    BibTeX
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    7: Bibliography L
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  • P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright Ο‰ function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
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    External Links:
    ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt
    Referenced by:
    §4.13, 2nd item, §4.48(iv).
    Encodings:
    BibTeX
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  • D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.
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    External Links:
    MathReview (D. H. Lehmer), ZentralBlatt
    Referenced by:
    §24.11.
    Encodings:
    BibTeX
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  • D. R. Lehman and J. S. O’Connell (1973) Graphical Recoupling of Angular Momenta. Technical report U.S. Government Printing Office, National Bureau of Standards, Washington, D.C..
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    Notes:
    Monograph 136
    Referenced by:
    §34.9.
    Encodings:
    BibTeX
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  • J. L. López (2000) Asymptotic expansions of symmetric standard elliptic integrals. SIAM J. Math. Anal. 31 (4), pp. 754–775.
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    External Links:
    ISSN 1095-7154, Document, MathReview (P. Anandani), ZentralBlatt
    Referenced by:
    §19.27(vi), §2.6(ii).
    Encodings:
    BibTeX
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  • D. W. Lozier (1980) Numerical Solution of Linear Difference Equations. NBSIR Technical Report 80-1976, National Bureau of Standards, Gaithersburg, MD 20899.
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    Notes:
    (Available from National Technical Information Service, Springfield, VA 22161.)
    Referenced by:
    §16.25, §3.6(vii).
    Encodings:
    BibTeX
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    8: Bibliography M
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  • A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.
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    Notes:
    Includes Fortran code, maximum accuracy 20S.
    External Links:
    ISSN 1017-1398, Document, MathReview, ZentralBlatt
    Referenced by:
    §6.13, 4th item, §6.21(ii).
    Encodings:
    BibTeX
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  • Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.
    β“˜
    External Links:
    ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
    Referenced by:
    §10.74(iii).
    Encodings:
    BibTeX
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  • J. C. P. Miller (1950) On the choice of standard solutions for a homogeneous linear differential equation of the second order. Quart. J. Mech. Appl. Math. 3 (2), pp. 225–235.
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    External Links:
    Document, MathReview (W. R. Wasow), ZentralBlatt
    Referenced by:
    §12.2(vi), §2.7(iv), §2.7(iv).
    Encodings:
    BibTeX
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  • J. C. P. Miller (1952) On the choice of standard solutions to Weber’s equation. Proc. Cambridge Philos. Soc. 48, pp. 428–435.
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    External Links:
    Document, MathReview (W. Wasow), ZentralBlatt
    Referenced by:
    §12.1, §12.14(i), §12.2(i).
    Encodings:
    BibTeX
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  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.
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    External Links:
    ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt
    Referenced by:
    §18.27(v).
    Encodings:
    BibTeX
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    9: Bibliography W
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  • P. L. Walker (2012) Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.
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    External Links:
    Document
    Referenced by:
    (20.7.34), §20.7(ix).
    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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  • G. Wolf (2008) On the asymptotic behavior of the Fourier coefficients of Mathieu functions. J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.
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    External Links:
    ISSN 1044-677X, FullText
    Referenced by:
    §28.4(vii).
    Encodings:
    BibTeX
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  • R. Wong and H. Y. Zhang (2007) Asymptotic solutions of a fourth order differential equation. Stud. Appl. Math. 118 (2), pp. 133–152.
    β“˜
    External Links:
    ISSN 0022-2526, Document, MathReview Entry
    Referenced by:
    §2.8(vi).
    Encodings:
    BibTeX
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    10: 28.1 Special Notation
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    ce ν ⁑ ( z , q ) , se ν ⁑ ( z , q ) , fe n ⁑ ( z , q ) , ge n ⁑ ( z , q ) , me ν ⁑ ( z , q ) ,
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    Abramowitz and Stegun (1964, Chapter 20)
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    National Bureau of Standards (1967)
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