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1—10 of 30 matching pages
1: 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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The third Appell function for one large variable.
J. Approx. Theory 165, pp. 60–69.
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The second Appell function for one large variable.
Mediterr. J. Math. 10 (4), pp. 1853–1865.
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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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2: 18.40 Methods of Computation
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►The problem of moments is simply stated and the early work of Stieltjes, Markov, and Chebyshev on this problem was the origin of the understanding of the importance of both continued fractions and OP’s in many areas of analysis.
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►See Gautschi (1983) for examples of numerically stable and unstable use of the above recursion relations, and how one can then usefully differentiate between numerical results of low and high precision, as produced thereby.
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►Results of low ( to decimal digits) precision for are easily obtained for to .
Gautschi (2004, p. 119–120) has explored the limit via the Wynn -algorithm, (3.9.11) to accelerate convergence, finding four to eight digits of precision in , depending smoothly on
, for , for an example involving first numerator Legendre OP’s.
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►This is a challenging case as the desired
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has an essential singularity at .
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3: 23.9 Laurent and Other Power Series
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►For , and with as in §23.3(i),
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23.9.6
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►Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as .
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4: 7.23 Tables
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§7.23(ii) Real Variables
… ►§7.23(iii) Complex Variables,
… ►Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of , , , 7D and 8D, respectively; the real and imaginary parts of , , , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.
Fettis et al. (1973) gives the first 100 zeros of and (the table on page 406 of this reference is for , not for ), 11S.
5: Bibliography O
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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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Note on backward recurrence algorithms.
Math. Comp. 26 (120), pp. 941–947.
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On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions.
J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 225–243.
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OEIS Foundation, Inc., Highland Park, New Jersey.
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Numerical Computing with IEEE Floating Point Arithmetic.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
6: 10.73 Physical Applications
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►and on separation of variables we obtain solutions of the form , from which a solution satisfying prescribed boundary conditions may be constructed.
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►on assuming a time dependence of the form .
…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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►On separation of variables into cylindrical coordinates, the Bessel functions , and modified Bessel functions and , all appear.
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►With the spherical harmonic defined as in §14.30(i), the solutions are of the form with , , , or , depending on the boundary conditions.
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7: Bibliography C
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A note on Euler numbers and polynomials.
Nagoya Math. J. 7, pp. 35–43.
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On computing elliptic integrals and functions.
J. Math. and Phys. 44, pp. 36–51.
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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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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8: 26.5 Lattice Paths: Catalan Numbers
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►It counts the number of lattice paths from to that stay on or above the line .
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26.5.2
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9: 6.16 Mathematical Applications
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6.16.1
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6.16.4
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►Similarly if , then the limiting value of undershoots by approximately 10%, and so on.
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6.16.5
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