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11: 5.13 Integrals
12: 34.5 Basic Properties: Symbol
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►For further recursion relations see Varshalovich et al. (1988, §9.6) and Edmonds (1974, pp. 98–99).
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►Equation (34.5.23) can be regarded as an alternative definition of the symbol.
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13: 27.2 Functions
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►Functions in this section derive their properties from the fundamental
theorem of arithmetic, which states that every integer can be represented uniquely as a product of prime powers,
…( is defined to be 0.)
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►It is the special case of the function that counts the number of ways of expressing as the product of factors, with the order of factors taken into account.
…Note that .
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►Table 27.2.2 tabulates the Euler totient function , the divisor function (), and the sum of the divisors (), for .
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14: 7.14 Integrals
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7.14.5
,
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7.14.7
,
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►For collections of integrals see Apelblat (1983, pp. 131–146), Erdélyi et al. (1954a, vol. 1, pp. 40, 96, 176–177), Geller and Ng (1971), Gradshteyn and Ryzhik (2000, §§5.4 and 6.28–6.32), Marichev (1983, pp. 184–189), Ng and Geller (1969), Oberhettinger (1974, pp. 138–139, 142–143), Oberhettinger (1990, pp. 48–52, 155–158), Oberhettinger and Badii (1973, pp. 171–172, 179–181), Prudnikov et al. (1986b, vol. 2, pp. 30–36, 93–143), Prudnikov et al. (1992a, §§3.7–3.8), and Prudnikov et al. (1992b, §§3.7–3.8).
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15: Bibliography F
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Tables of Elliptic Integrals of the First, Second, and Third Kind.
Technical report
Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
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Algorithm 309. Gamma function with arbitrary precision.
Comm. ACM 10 (8), pp. 511–512.
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Diffraction of radio waves around the earth’s surface.
Acad. Sci. USSR. J. Phys. 9, pp. 255–266.
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Travel time surface of a transverse cusp caustic produced by reflection of acoustical transients from a curved metal surface.
J. Acoust. Soc. Amer. 95 (2), pp. 650–660.
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Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions.
In Progress in Optics, E. Wolf (Ed.),
Vol. 9, pp. 311–407.
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16: 9.8 Modulus and Phase
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►Primes denote differentiations with respect to , which is continued to be assumed real and nonpositive.
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9.8.20
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9.8.21
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9.8.22
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9.8.23
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17: Bibliography B
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The generating function of Jacobi polynomials.
J. London Math. Soc. 13, pp. 8–12.
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Bäcklund transformations and solution hierarchies for the fourth Painlevé equation.
Stud. Appl. Math. 95 (1), pp. 1–71.
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Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces.
J. Phys. A 8 (4), pp. 566–584.
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Asymptotic behavior of the Pollaczek polynomials and their zeros.
Stud. Appl. Math. 96, pp. 307–338.
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Irregular primes and cyclotomic invariants to 12 million.
J. Symbolic Comput. 31 (1-2), pp. 89–96.
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18: 4.40 Integrals
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►Throughout this section the variables are assumed to be real.
The results in §§4.40(ii) and 4.40(iv) can be extended to the complex plane by using continuous branches and avoiding singularities.
…
►Extensive compendia of indefinite and definite integrals of hyperbolic functions include Apelblat (1983, pp. 96–109), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 139–160), Gröbner and Hofreiter (1950, pp. 160–167), Gradshteyn and Ryzhik (2000, Chapters 2–4), and Prudnikov et al. (1986a, §§1.4, 1.8, 2.4, 2.8).
19: 9.9 Zeros
20: Bibliography J
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Further results on the computation of incomplete gamma functions.
In Analytic Theory of Continued Fractions, II
(Pitlochry/Aviemore, 1985), W. J. Thron (Ed.),
Lecture Notes in Math. 1199, pp. 67–89.
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Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent.
Phys. D 1 (1), pp. 80–158.
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Asymptotic formulas for the zeros of the Meixner polynomials.
J. Approx. Theory 96 (2), pp. 281–300.
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Modifications of Coulombic interactions by polarizable atoms.
Math. Proc. Cambridge Philos. Soc. 80 (3), pp. 535–539.
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Memoire sur l’itération des fonctions rationnelles.
J. Math. Pures Appl. 8 (1), pp. 47–245 (French).