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1: 27.2 Functions
Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer n > 1 can be represented uniquely as a product of prime powers, …Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … This is the number of positive integers n that are relatively prime to n ; ϕ ( n ) is Euler’s totient. If ( a , n ) = 1 , then the Euler–Fermat theorem states that …It is the special case k = 2 of the function d k ( n ) that counts the number of ways of expressing n as the product of k factors, with the order of factors taken into account. …
2: Bibliography L
  • P. W. Lawrence, R. M. Corless, and D. J. Jeffrey (2012) Algorithm 917: complex double-precision evaluation of the Wright ω function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
  • D. J. Leeming (1977) An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.
  • D. R. Lehman, W. C. Parke, and L. C. Maximon (1981) Numerical evaluation of integrals containing a spherical Bessel function by product integration. J. Math. Phys. 22 (7), pp. 1399–1413.
  • P. Linz and T. E. Kropp (1973) A note on the computation of integrals involving products of trigonometric and Bessel functions. Math. Comp. 27 (124), pp. 871–872.
  • S. K. Lucas (1995) Evaluating infinite integrals involving products of Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 269–282.
  • 3: Bibliography
  • M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
  • T. Agoh and K. Dilcher (2011) Integrals of products of Bernoulli polynomials. J. Math. Anal. Appl. 381 (1), pp. 10–16.
  • J. R. Albright (1977) Integrals of products of Airy functions. J. Phys. A 10 (4), pp. 485–490.
  • R. Askey, T. H. Koornwinder, and M. Rahman (1986) An integral of products of ultraspherical functions and a q -extension. J. London Math. Soc. (2) 33 (1), pp. 133–148.
  • 4: Bibliography W
  • S. S. Wagstaff (2002) Prime Divisors of the Bernoulli and Euler Numbers. In Number Theory for the Millennium, III (Urbana, IL, 2000), pp. 357–374.
  • P. L. Walker (2007) The zeros of Euler’s psi function and its derivatives. J. Math. Anal. Appl. 332 (1), pp. 607–616.
  • P. L. Walker (2012) Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.
  • R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.
  • J. Wishart (1928) The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A, pp. 32–52.
  • 5: 12.10 Uniform Asymptotic Expansions for Large Parameter
    and the coefficients γ s are defined by …
    γ 0 = 1 ,
    where h ( μ ) and γ s are as in §12.10(ii). … and the coefficients 𝖠 s ( τ ) are the product of τ s and a polynomial in τ of degree 2 s . …
    𝖠 1 ( τ ) = 1 12 τ ( 20 τ 2 + 30 τ + 9 ) ,
    6: Bibliography D
  • K. Dilcher (1996) Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.
  • R. McD. Dodds and G. Wiechers (1972) Vector coupling coefficients as products of prime factors. Comput. Phys. Comm. 4 (2), pp. 268–274.
  • B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.
  • T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.
  • L. Durand (1978) Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal. 9 (1), pp. 76–86.