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11: Mathematical Introduction
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►The NIST Handbook has essentially the same objective as the Handbook of Mathematical Functions that was issued in 1964 by the National Bureau of Standards as Number 55 in the NBS Applied Mathematics Series (AMS).
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►As a consequence, in addition to providing more information about the special functions that were covered in AMS 55, the NIST Handbook includes several special functions that have appeared in the interim in applied mathematics, the physical sciences, and engineering, as well as in other areas.
See, for example, Chapters 16, 17, 18, 19, 21, 27, 29, 31, 32, 34, 35, and 36.
►Two other ways in which this Handbook differs from AMS 55, and other handbooks, are as follows.
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►For equations or other technical information that appeared previously in AMS 55, the DLMF usually includes the corresponding AMS 55 equation number, or other form of reference, together with corrections, if needed.
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12: 6.14 Integrals
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►For collections of integrals, see Apelblat (1983, pp. 110–123), Bierens de Haan (1939, pp. 373–374, 409, 479, 571–572, 637, 664–673, 680–682, 685–697), Erdélyi et al. (1954a, vol. 1, pp. 40–42, 96–98, 177–178, 325), Geller and Ng (1969), Gradshteyn and Ryzhik (2000, §§5.2–5.3 and 6.2–6.27), Marichev (1983, pp. 182–184), Nielsen (1906b), Oberhettinger (1974, pp. 139–141), Oberhettinger (1990, pp. 53–55 and 158–160), Oberhettinger and Badii (1973, pp. 172–179), Prudnikov et al. (1986b, vol. 2, pp. 24–29 and 64–92), Prudnikov et al. (1992a, §§3.4–3.6), Prudnikov et al. (1992b, §§3.4–3.6), and Watrasiewicz (1967).
13: William P. Reinhardt
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►Older work on the scattering theory of the atomic Coulomb problem led to the discovery of new classes of orthogonal polynomials relating to the spectral theory of Schrödinger operators, and new uses of old ones: this work was strongly motivated by his original ownership of a 1964 hard copy printing of the original AMS 55 NBS Handbook of Mathematical Functions.
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14: Bibliography H
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Algorithm 55: Complete elliptic integral of the first kind.
Comm. ACM 4 (4), pp. 180.
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Inverse virial symmetry of diatomic potential curves.
J. Chem. Phys. 109 (1), pp. 11–19.
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Error bounds for asymptotic approximations of zeros of Hankel functions occurring in diffraction problems.
J. Mathematical Phys. 11 (8), pp. 2501–2504.
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Algorithm 395: Student’s t-distribution.
Comm. ACM 13 (10), pp. 617–619.
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Two Parametric Eigenvalue Problems of Differential Equations.
In Spectral Theory of Differential Operators (Birmingham, AL,
1981),
North-Holland Math. Stud., Vol. 55, pp. 233–241.
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15: 28.6 Expansions for Small
16: 24.2 Definitions and Generating Functions
17: Bibliography E
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The penetration of a potential barrier by electrons.
Phys. Rev. 35 (11), pp. 1303–1309.
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Higher Transcendental Functions. Vol. II.
McGraw-Hill Book Company, Inc., New York-Toronto-London.
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Painlevé transcendent describes quantum correlation function of the antiferromagnet away from the free-fermion point.
J. Phys. A 29 (17), pp. 5619–5626.
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Institutiones Calculi Integralis.
Opera Omnia (1), Vol. 11, pp. 110–113.
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A comparison of some methods for the evaluation of highly oscillatory integrals.
J. Comput. Appl. Math. 112 (1-2), pp. 55–69.
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18: Bibliography F
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Asymptotics of the spectrum of the Heun equation and of Heun functions.
Izv. Akad. Nauk SSSR Ser. Mat. 55 (3), pp. 631–646 (Russian).
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Polynomial relations in the Heisenberg algebra.
J. Math. Phys. 35 (11), pp. 6144–6149.
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On a unified approach to transformations and elementary solutions of Painlevé equations.
J. Math. Phys. 23 (11), pp. 2033–2042.
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Application of the -function theory of Painlevé equations to random matrices: , , the LUE, JUE, and CUE.
Comm. Pure Appl. Math. 55 (6), pp. 679–727.
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Algorithm 435: Modified incomplete gamma function.
Comm. ACM 15 (11), pp. 993–995.
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19: 3.9 Acceleration of Convergence
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Table 3.9.1: Shanks’ transformation for .
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►For examples and other transformations for convergent sequences and series, see Wimp (1981, pp. 156–199), Brezinski and Redivo Zaglia (1991, pp. 55–72), and Sidi (2003, Chapters 6, 12–13, 15–16, 19–24, and pp. 483–492).
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2 | 0.82111 11111 11 | 0.82243 44785 14 | 0.82246 61821 45 | 0.82246 70102 48 | 0.82246 70327 79 |
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