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11—20 of 21 matching pages
11: Bibliography P
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An asymptotic formula for the incomplete gamma function.
Ε½. VyΔisl. Mat. i Mat. Fiz. 5, pp. 118–121 (Russian).
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An inequality for the Bessel function
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SIAM J. Math. Anal. 15 (1), pp. 203–205.
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The Stokes phenomenon associated with the Hurwitz zeta function
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Proc. Roy. Soc. London Ser. A 461, pp. 297–304.
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Complex Interval Arithmetic and its Applications.
Mathematical Research, Vol. 105, Wiley-VCH Verlag Berlin GmbH, Berlin.
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Automatic computation of Bessel function integrals.
Comput. Phys. Comm. 25 (3), pp. 289–295.
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12: Bibliography I
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IEEE Standard for Interval Arithmetic: IEEE Std 1788-2015.
The Institute of Electrical and Electronics Engineers, Inc..
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IEEE Standard for Interval Arithmetic: IEEE Std 1788.1-2017.
The Institute of Electrical and Electronics Engineers, Inc..
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An integral which occurs in statistics.
Proceedings of the Cambridge Philosophical Society 29, pp. 271–276.
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The real roots of Bernoulli polynomials.
Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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An electrostatics model for zeros of general orthogonal polynomials.
Pacific J. Math. 193 (2), pp. 355–369.
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13: 18.39 Applications in the Physical Sciences
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An Introductory Remark
… βΊwhere the orthogonality measure is now , … βΊOrthogonality, with measure for , for fixed … βΊnormalized with measure , . … βΊwhich maps onto . …14: 36.5 Stokes Sets
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βΊStokes sets are surfaces (codimension one) in space, across which or acquires an exponentially-small asymptotic contribution (in ), associated with a complex critical point of or .
…where denotes a real critical point (36.4.1) or (36.4.2), and denotes a critical point with complex or , connected with by a steepest-descent path (that is, a path where ) in complex or space.
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36.5.4
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36.5.7
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15: 12.10 Uniform Asymptotic Expansions for Large Parameter
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βΊWith the upper sign in (12.10.2), expansions can be constructed for large in terms of elementary functions that are uniform for (§2.8(ii)).
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βΊThroughout this section the symbol again denotes an arbitrary small positive constant.
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βΊuniformly for , where
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βΊAn example is the following modification of (12.10.3)
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16: 5.11 Asymptotic Expansions
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5.11.3
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βΊWrench (1968) gives exact values of up to .
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5.11.8
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βΊFor error bounds for (5.11.8) and an exponentially-improved extension, see Nemes (2013b).
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17: Bibliography C
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An algorithm for the Fourier-Bessel transform.
Comput. Phys. Comm. 23 (4), pp. 343–353.
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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An extension of a Kummer’s quadratic transformation formula with an application.
Proc. Jangjeon Math. Soc. 16 (2), pp. 229–235.
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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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18: Bibliography F
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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Tables of Elliptic Integrals of the First, Second, and Third Kind.
Technical report
Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
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Tables of the line broadening function
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Monthly Notices Roy. Astronom. Soc. 129, pp. 221–235.
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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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19: 28.1 Special Notation
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integers. | |
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order of the Mathieu function or modified Mathieu function. (When is an integer it is often replaced by .) | |
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Abramowitz and Stegun (1964, Chapter 20)
…20: 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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Introduction to Interval Computations.
Computer Science and Applied Mathematics, Academic Press Inc., New York.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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An integral for Jacobi polynomials.
Simon Stevin 46, pp. 165–169.
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