central%20differences%20in%20imaginary%20direction
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1: Gergő Nemes
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► 1988 in Szeged, Hungary) is a Research Fellow at the Alfréd Rényi Institute of Mathematics in Budapest, Hungary.
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► in mathematics (with distinction) and a M.
…in mathematics (with honours) from Loránd Eötvös University, Budapest, Hungary and a Ph.
… in mathematics from Central European University in Budapest, Hungary.
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►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions.
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2: Staff
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William P. Reinhardt, University of Washington, Chaps. 20, 22, 23
Peter L. Walker, American University of Sharjah, Chaps. 20, 22, 23
Mourad E. H. Ismail, University of Central Florida
William P. Reinhardt, University of Washington, for Chaps. 20, 22, 23
Peter L. Walker, American University of Sharjah, for Chaps. 20, 22, 23
3: 20 Theta Functions
Chapter 20 Theta Functions
…4: 18.40 Methods of Computation
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►Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree.
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►A simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let be a positive integer and define
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►in which
…Results of low ( to decimal digits) precision for are easily obtained for to .
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5: 18.1 Notation
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-Differences
►Forward differences: … ►Backward differences: … ►Central differences in imaginary direction: … ►In Koekoek et al. (2010) denotes the operator .6: Mourad E. H. Ismail
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► 1944, in Cairo, Egypt) is a Distinguished Research Professor in the Department of Mathematics of the University of Central Florida.
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►His well-known book Classical and Quantum Orthogonal Polynomials in One Variable was published by Cambridge University Press in 2005 and reprinted with corrections in paperback in Ismail (2009).
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► Koelink), Developments in Mathematics, v.
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►Ismail was elected a fellow of the American Mathematical Society in 2014.
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7: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
Pagurova (1963) tabulates and (with different notation) for , to 7D.
Abramowitz and Stegun (1964, pp. 245–248) tabulates for , to 7D; also for , to 6S.
Pagurova (1961) tabulates for , to 4-9S; for , to 7D; for , to 7S or 7D.
Zhang and Jin (1996, Table 19.1) tabulates for , to 7D or 8S.
8: 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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The central two-point connection problem for the Heun class of ODEs.
J. Phys. A 31 (18), pp. 4249–4261.
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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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Functions on a parabolic cylinder with a negative integer index.
Differ. Uravn. 21 (11), pp. 2001–2003, 2024 (Russian).
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The second Painlevé equation.
Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).
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9: Bibliography W
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Asymptotic expansions for second-order linear difference equations with a turning point.
Numer. Math. 94 (1), pp. 147–194.
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Linear difference equations with transition points.
Math. Comp. 74 (250), pp. 629–653.
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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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On the central connection problem for the double confluent Heun equation.
Math. Nachr. 195, pp. 267–276.
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Asymptotic expansions for second-order linear difference equations. II.
Stud. Appl. Math. 87 (4), pp. 289–324.
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10: 10.73 Physical Applications
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►Laplace’s equation governs problems in heat conduction, in the distribution of potential in an electrostatic field, and in hydrodynamics in the irrotational motion of an incompressible fluid.
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►Consequently, Bessel functions , and modified Bessel functions , are central to the analysis of microwave and optical transmission in waveguides, including coaxial and fiber.
See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25).
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►More recently, Bessel functions appear in the inverse problem in wave propagation, with applications in medicine, astronomy, and acoustic imaging.
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►In quantum mechanics the spherical Bessel functions arise in the solution of the Schrödinger wave equation for a particle in a central potential.
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