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11: 6.7 Integral Representations
§6.7 Integral Representations
§6.7(ii) Sine and Cosine Integrals
§6.7(iii) Auxiliary Functions
For collections of integral representations see Bierens de Haan (1939, pp. 56–59, 72–73, 82–84, 121, 133–136, 155, 179–181, 223, 225–227, 230, 259–260, 374, 377, 397–398, 408, 416, 424, 431, 438–439, 442–444, 488, 496–500, 567–571, 585, 602, 638, 675–677), Corrington (1961), Erdélyi et al. (1954a, vol. 1, pp. 267–270), Geller and Ng (1969), Nielsen (1906b), Oberhettinger (1974, pp. 244–246), Oberhettinger and Badii (1973, pp. 364–371), and Watrasiewicz (1967).
12: 7.7 Integral Representations
§7.7 Integral Representations
§7.7(i) Error Functions and Dawson’s Integral
§7.7(ii) Auxiliary Functions
Mellin–Barnes Integrals
For other integral representations see Erdélyi et al. (1954a, vol. 1, pp. 265–267, 270), Ng and Geller (1969), Oberhettinger (1974, pp. 246–248), and Oberhettinger and Badii (1973, pp. 371–377).
13: 14.26 Uniform Asymptotic Expansions
See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.
14: 5.14 Multidimensional Integrals
§5.14 Multidimensional Integrals
5.14.7 1 ( 2 π ) n [ - π , π ] n 1 j < k n | e i θ j - e i θ k | 2 b d θ 1 d θ n = Γ ( 1 + b n ) ( Γ ( 1 + b ) ) n , b > - 1 / n .
15: 9.17 Methods of Computation
§9.17(iii) Integral Representations
Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing Ai ( z ) in the complex plane, once values of this function can be generated on the positive real axis. …
16: 35.10 Methods of Computation
Other methods include numerical quadrature applied to double and multiple integral representations. …
17: 14.32 Methods of Computation
  • Quadrature (§3.5) of the integral representations given in §§14.12, 14.19(iii), 14.20(iv), and 14.25; see Segura and Gil (1999) and Gil et al. (2000).

  • 18: 10.54 Integral Representations
    §10.54 Integral Representations
    Additional integral representations can be obtained by combining the definitions (10.47.3)–(10.47.9) with the results given in §10.9 and §10.32.
    19: 8.6 Integral Representations
    §8.6 Integral Representations
    §8.6(i) Integrals Along the Real Line
    §8.6(ii) Contour Integrals
    Mellin–Barnes Integrals
    §8.6(iii) Compendia
    20: 11.5 Integral Representations
    §11.5 Integral Representations
    §11.5(i) Integrals Along the Real Line
    §11.5(ii) Contour Integrals
    Mellin–Barnes Integrals
    §11.5(iii) Compendia