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11—20 of 951 matching pages
11: 5.13 Integrals
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►In (5.13.1) the integration path is a straight line parallel to the imaginary axis.
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5.13.1
, ,
.
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5.13.2
, .
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5.13.4
.
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5.13.5
, .
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12: 27.2 Functions
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►Such a set is a reduced
residue system modulo .
…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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►where is a prime power with ; otherwise .
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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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13: 34.5 Basic Properties: Symbol
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►If any lower argument in a
symbol is , , or , then the symbol has a simple algebraic form.
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34.5.1
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34.5.2
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34.5.3
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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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14: 7.14 Integrals
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►When the limit is taken.
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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).
In a series of ten papers Hadži (1968, 1969, 1970, 1972, 1973, 1975a, 1975b, 1976a, 1976b, 1978) gives many integrals containing error functions and Fresnel integrals, also in combination with the hypergeometric function, confluent hypergeometric functions, and generalized hypergeometric functions.
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
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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Asymptotic behavior of the Pollaczek polynomials and their zeros.
Stud. Appl. Math. 96, pp. 307–338.
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Algorithm 21: Bessel function for a set of integer orders.
Comm. ACM 3 (11), pp. 600.
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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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4.40.7
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4.40.8
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4.40.9
,
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4.40.10
, .
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►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: Bibliography J
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Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles.
Eurandom Preprint Series
Technical Report 14, Eurandom, Eindhoven, The Netherlands.
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A note on sampling expansion for a transform with parabolic cylinder kernel.
Inform. Sci. 26 (2), pp. 155–158.
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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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Introduction to Asymptotics: A Treatment Using Nonstandard Analysis.
World Scientific Publishing Co. Inc., River Edge, NJ.
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Memoire sur l’itération des fonctions rationnelles.
J. Math. Pures Appl. 8 (1), pp. 47–245 (French).
20: 9.9 Zeros
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►They are denoted by , , , , respectively, arranged in ascending order of absolute value for
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►They lie in the sectors and , and are denoted by , , respectively, in the former sector, and by , , in the conjugate sector, again arranged in ascending order of absolute value (modulus) for See §9.3(ii) for visualizations.
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►If is regarded as a continuous variable, then
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9.9.21
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►For error bounds for the asymptotic expansions of , , , and see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999).
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