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1: 9.9 Zeros
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βΊThey are denoted by , , , , respectively, arranged in ascending order of absolute value for
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βΊIf is regarded as a continuous variable, then
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βΊFor large
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9.9.6
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βΊFor error bounds for the asymptotic expansions of , , , and see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999).
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2: Bibliography C
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Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial as the index and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials.
Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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On the representation of a large even integer as the sum of a prime and the product of at most two primes.
Kexue Tongbao (Foreign Lang. Ed.) 17, pp. 385–386.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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Coulomb effects in the Klein-Gordon equation for pions.
Phys. Rev. C 20 (2), pp. 696–704.
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3: 27.2 Functions
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βΊEuclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
…They tend to thin out among the large integers, but this thinning out is not completely regular.
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βΊand if is the smallest positive integer such that , then is a primitive root mod .
…Such a set is a reduced
residue system modulo .
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βΊwhere is a prime power with ; otherwise .
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4: 27.15 Chinese Remainder Theorem
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βΊThe Chinese remainder theorem states that a system of congruences , always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod ), where is the product of the moduli.
βΊThis theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation.
For example, suppose a lengthy calculation involves many 10-digit integers.
…Their product has 20 digits, twice the number of digits in the data.
…These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result , which is correct to 20 digits.
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5: Bibliography N
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Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes.
ACM Trans. Math. Software 18 (3), pp. 345–349.
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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Coulomb Functions for Large Values of the Parameter
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Technical report
Atomic Energy of Canada Limited, Chalk
River, Ontario.
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A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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6: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann.
Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter.
SIAM J. Math. Anal. 21 (4), pp. 995–1018.
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Conical functions with one or both parameters large.
Proc. Roy. Soc. Edinburgh Sect. A 119 (3-4), pp. 311–327.
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Uniform asymptotic expansions for associated Legendre functions of large order.
Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.
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7: Bibliography B
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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Anharmonic oscillator. II. A study of perturbation theory in large order.
Phys. Rev. D 7, pp. 1620–1636.
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A short table of the functions , from to
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Phil. Mag. Series 7 20, pp. 343–347.
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Coulomb functions for large charges and small velocities.
Phys. Rev. (2) 97 (2), pp. 542–554.
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Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation.
J. Phys. A 30 (2), pp. 559–571.
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8: 28.35 Tables
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National Bureau of Standards (1967) includes the eigenvalues , for with , and with ; Fourier coefficients for and for , , respectively, and various values of in the interval ; joining factors , for with (but in a different notation). Also, eigenvalues for large values of . Precision is generally 8D.
9: 3.8 Nonlinear Equations
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βΊfor all sufficiently large, where and are independent of , then the sequence is said to have convergence of the
th order.
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βΊFor moderate or large values of it is not uncommon for the magnitude of the right-hand side of (3.8.14) to be very large compared with unity, signifying that the computation of zeros of polynomials is often an ill-posed problem.
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βΊConsider and .
We have and .
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βΊIt is called a Julia set.
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10: Bibliography F
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Uniform asymptotic expansions for hypergeometric functions with large parameters IV.
Anal. Appl. (Singap.) 12 (6), pp. 667–710.
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Uniform asymptotic expansions of certain classes of Meijer -functions for a large parameter.
SIAM J. Math. Anal. 4 (3), pp. 482–507.
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Uniform asymptotic expansions of a class of Meijer -functions for a large parameter.
SIAM J. Math. Anal. 14 (6), pp. 1204–1253.
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On weighted polynomial approximation on the whole real axis.
Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
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