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21: Bibliography O
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Fourier Expansions. A Collection of Formulas.
Academic Press, New York-London.
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On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles.
Methods Appl. Anal. 7 (4), pp. 727–745.
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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series.
J. Inst. Math. Appl. 20 (3), pp. 379–391.
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Error bounds for asymptotic expansions in turning-point problems.
J. Soc. Indust. Appl. Math. 12 (1), pp. 200–214.
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Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations.
Methods Appl. Anal. 1 (1), pp. 1–13.
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22: 2.11 Remainder Terms; Stokes Phenomenon
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►Secondly, the asymptotic series represents an infinite class of functions, and the remainder depends on which member we have in mind.
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§2.11(iii) Exponentially-Improved Expansions
… ►For illustration, we give re-expansions of the remainder terms in the expansions (2.7.8) arising in differential-equation theory. … ►In this way we arrive at hyperasymptotic expansions. … ►For example, using double precision is found to agree with (2.11.31) to 13D. …23: 36 Integrals with Coalescing Saddles
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24: 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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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
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Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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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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Error analysis in a uniform asymptotic expansion for the generalised exponential integral.
J. Comput. Appl. Math. 80 (1), pp. 127–161.
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25: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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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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Coulomb functions (negative energies).
Comput. Phys. Comm. 20 (3), pp. 447–458.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Bessel functions and modular relations of higher type and hyperbolic differential equations.
Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.
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26: 5.11 Asymptotic Expansions
§5.11 Asymptotic Expansions
►§5.11(i) Poincaré-Type Expansions
… ►Wrench (1968) gives exact values of up to . … ►For re-expansions of the remainder terms in (5.11.1) and (5.11.3) in series of incomplete gamma functions with exponential improvement (§2.11(iii)) in the asymptotic expansions, see Berry (1991), Boyd (1994), and Paris and Kaminski (2001, §6.4). … ►27: 9.9 Zeros
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►They lie in the sectors and , and are denoted by , , respectively, in the former sector, and by , , in the conjugate sector, again arranged in ascending order of absolute value (modulus) for See §9.3(ii) for visualizations.
►For the distribution in
of the zeros of , where is an arbitrary complex constant, see Muraveĭ (1976) and Gil and Segura (2014).
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§9.9(iv) Asymptotic Expansions
… ►For error bounds for the asymptotic expansions of , , , and see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). … ►Tables 9.9.3 and 9.9.4 give the corresponding results for the first ten complex zeros of and in the upper half plane. …28: Peter L. Walker
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► 1942 in Dorchester, U.
…He began his academic career in 1964 at the University of Lancaster, U.
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►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004.
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►Walker is now retired and living in Cheltenham, UK.
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29: 12.11 Zeros
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►Lastly, when , (Hermite polynomial case) has zeros and they lie in the interval .
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