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1: 16 Generalized Hypergeometric Functions & Meijer G-Function
Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function
2: 27.15 Chinese Remainder Theorem
Choose four relatively prime moduli m 1 , m 2 , m 3 , and m 4 of five digits each, for example 2 16 3 , 2 16 1 , 2 16 + 1 , and 2 16 + 3 . …
3: 20 Theta Functions
4: Adri B. Olde Daalhuis
5: 32.12 Asymptotic Approximations for Complex Variables
See Fokas et al. (2006, Chapter 16). …
6: 27.22 Software
  • Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below 10 16 . Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard p 1 , and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

    For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

  • 7: Staff
  • Richard A. Askey, University of Wisconsin, Chaps. 1, 5, 16

  • Adri B. Olde Daalhuis, University of Edinburgh, Chaps. 13, 15, 16

  • Adri B. Olde Daalhuis, The University of Edinburgh, for Chaps. 13, 15, 16

  • 8: 29.7 Asymptotic Expansions
    29.7.7 τ 3 = p 2 14 ( ( 1 + k 2 ) 4 ( 33 p 4 + 410 p 2 + 405 ) 24 k 2 ( 1 + k 2 ) 2 ( 7 p 4 + 90 p 2 + 95 ) + 16 k 4 ( 9 p 4 + 130 p 2 + 173 ) ) ,
    29.7.8 τ 4 = 1 2 16 ( ( 1 + k 2 ) 5 ( 63 p 6 + 1260 p 4 + 2943 p 2 + 486 ) 8 k 2 ( 1 + k 2 ) 3 ( 49 p 6 + 1010 p 4 + 2493 p 2 + 432 ) + 16 k 4 ( 1 + k 2 ) ( 35 p 6 + 760 p 4 + 2043 p 2 + 378 ) ) .
    9: 15.3 Graphics
    See accompanying text
    Figure 15.3.1: F ( 4 3 , 9 16 ; 14 5 ; x ) , 100 x 1 . Magnify
    See accompanying text
    Figure 15.3.5: F ( 4 3 , 9 16 ; 14 5 ; x + i y ) , 0 x 2 , 0.5 y 0.5 . … Magnify 3D Help
    10: 2.2 Transcendental Equations
    Conditions for the validity of the reversion process in are derived in Olver (1997b, pp. 14–16). …