Euler%20product
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1: 27.2 Functions
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►Functions in this section derive their properties from the fundamental
theorem of arithmetic, which states that every integer 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.
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►This is the number of positive integers that are relatively prime to ; is Euler’s totient.
►If , then the Euler–Fermat theorem states that
…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.
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2: Bibliography L
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Algorithm 917: complex double-precision evaluation of the Wright function.
ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.
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An asymptotic estimate for the Bernoulli and Euler numbers.
Canad. Math. Bull. 20 (1), pp. 109–111.
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Numerical evaluation of integrals containing a spherical Bessel function by product integration.
J. Math. Phys. 22 (7), pp. 1399–1413.
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A note on the computation of integrals involving products of trigonometric and Bessel functions.
Math. Comp. 27 (124), pp. 871–872.
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Evaluating infinite integrals involving products of Bessel functions of arbitrary order.
J. Comput. Appl. Math. 64 (3), pp. 269–282.
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3: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Integrals of products of Bernoulli polynomials.
J. Math. Anal. Appl. 381 (1), pp. 10–16.
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Integrals of products of Airy functions.
J. Phys. A 10 (4), pp. 485–490.
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An integral of products of ultraspherical functions and a -extension.
J. London Math. Soc. (2) 33 (1), pp. 133–148.
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4: Bibliography W
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Prime Divisors of the Bernoulli and Euler Numbers.
In Number Theory for the Millennium, III (Urbana, IL, 2000),
pp. 357–374.
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The zeros of Euler’s psi function and its derivatives.
J. Math. Anal. Appl. 332 (1), pp. 607–616.
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Reduction formulae for products of theta functions.
J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.
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The Nahm equations, finite-gap potentials and Lamé functions.
J. Phys. A 20 (10), pp. 2679–2683.
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The generalised product moment distribution in samples from a normal multivariate population.
Biometrika 20A, pp. 32–52.
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5: 12.10 Uniform Asymptotic Expansions for Large Parameter
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►and the coefficients are defined by
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►where and are as in §12.10(ii).
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►and the coefficients are the product of and a polynomial in of degree .
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6: Bibliography D
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Sums of products of Bernoulli numbers.
J. Number Theory 60 (1), pp. 23–41.
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Vector coupling coefficients as products of prime factors.
Comput. Phys. Comm. 4 (2), pp. 268–274.
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results.
SIAM J. Math. Anal. 9 (1), pp. 76–86.
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