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1: 2.11 Remainder Terms; Stokes Phenomenon
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►In the transition through , changes very rapidly, but smoothly, from one form to the other; compare the graph of its modulus in Figure 2.11.1 in the case .
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►Expansions similar to (2.11.15) can be constructed for many other special functions.
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►For other examples see Boyd (1990b), Paris (1992a, b), and Wong and Zhao (2002b).
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►For example, using double precision is found to agree with (2.11.31) to 13D.
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►Their extrapolation is based on assumed forms of remainder terms that may not always be appropriate for asymptotic expansions.
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2: 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
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…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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►Bessel functions enter in the study of the scattering of light and other electromagnetic radiation, not only from cylindrical surfaces but also in the statistical analysis involved in scattering from rough surfaces.
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►The McLachlan reference also includes other applications of Kelvin functions.
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3: 25.6 Integer Arguments
4: 27.2 Functions
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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►An equivalent form states that the th prime (when the primes are listed in increasing order) is asymptotic to as :
…Other examples of number-theoretic functions treated in this chapter are as follows.
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5: 23.9 Laurent and Other Power Series
§23.9 Laurent and Other Power Series
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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean.
SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
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On integral representations for Lamé and other special functions.
SIAM J. Math. Anal. 11 (4), pp. 702–723.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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A Maple package for symmetric functions.
J. Symbolic Comput. 20 (5-6), pp. 755–768.
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Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
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7: Bibliography K
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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The Scattering of Light and Other Electromagnetic Radiation.
Academic Press, New York.
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A vortex filament moving without change of form.
J. Fluid Mech. 112, pp. 397–409.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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8: 5.11 Asymptotic Expansions
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►Wrench (1968) gives exact values of up to .
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►For further information see Olver (1997b, pp. 293–295), and for other error bounds see Whittaker and Watson (1927, §12.33), Spira (1971), and Schäfke and Finsterer (1990).
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9: 18.39 Applications in the Physical Sciences
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►The fundamental quantum Schrödinger operator, also called the Hamiltonian, , is a second order differential operator of the form
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►The orthonormal stationary states and corresponding eigenvalues are then of the form
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