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1: 34.6 Definition: Symbol
§34.6 Definition: Symbol
βΊThe symbol may be defined either in terms of symbols or equivalently in terms of symbols: βΊ
34.6.1
βΊ
34.6.2
βΊThe symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments.
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2: 26.5 Lattice Paths: Catalan Numbers
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βΊ
is the Catalan number.
…(Sixty-six equivalent definitions of are given in Stanley (1999, pp. 219–229).)
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βΊ
26.5.3
βΊ
26.5.4
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βΊ
26.5.7
3: 6.14 Integrals
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βΊ
6.14.1
,
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βΊ
6.14.4
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βΊFor 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).
4: 5.13 Integrals
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5: 9.8 Modulus and Phase
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βΊ
9.8.20
βΊ
9.8.21
βΊ
9.8.22
βΊ
9.8.23
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βΊAlso, approximate values (25S) of the coefficients of the powers , , , are available in Sherry (1959).
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6: 33.20 Expansions for Small
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βΊwhere
βΊ
33.20.4
,
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βΊThe functions and are as in §§10.2(ii), 10.25(ii), and the coefficients are given by , , and
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βΊwhere is given by (33.14.11), (33.14.12), and
…The functions and are as in §§10.2(ii), 10.25(ii), and the coefficients are given by (33.20.6).
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7: 7.14 Integrals
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βΊ
7.14.1
, .
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βΊ
7.14.5
,
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βΊ
7.14.7
,
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βΊFor 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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8: 4.40 Integrals
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βΊExtensive 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).
9: Bibliography H
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βΊ
Certain integrals that contain a probability function.
Bul. Akad. Ε tiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).
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βΊ
Expansions for the probability function in series of ΔebyΕ‘ev polynomials and Bessel functions.
Bul. Akad. Ε tiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).
βΊ
Integrals that contain a probability function of complicated arguments.
Bul. Akad. Ε tiince RSS Moldoven. 1976 (1), pp. 80–84, 96 (Russian).
βΊ
Sums with cylindrical functions that reduce to the probability function and to related functions.
Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).
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βΊ
Algorithm 56: Complete elliptic integral of the second kind.
Comm. ACM 4 (4), pp. 180–181.
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10: 27.2 Functions
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βΊ
27.2.9
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βΊIt 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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βΊTable 27.2.2 tabulates the Euler totient function , the divisor function (), and the sum of the divisors (), for .
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βΊ