Debye expansions
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1: 8.22 Mathematical Applications
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►plays a fundamental role in re-expansions of remainder terms in asymptotic expansions, including exponentially-improved expansions and a smooth interpretation of the Stokes phenomenon.
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►See Paris and Cang (1997).
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►The Debye functions
and are closely related to the incomplete Riemann zeta function and the Riemann zeta function.
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2: 10.19 Asymptotic Expansions for Large Order
§10.19 Asymptotic Expansions for Large Order
►§10.19(i) Asymptotic Forms
… ►§10.19(ii) Debye’s Expansions
… ►For error bounds for the first of (10.19.6) see Olver (1997b, p. 382). … ►See also §10.20(i).3: 16.22 Asymptotic Expansions
§16.22 Asymptotic Expansions
►Asymptotic expansions of for large are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). For asymptotic expansions of Meijer -functions with large parameters see Fields (1973, 1983).4: 20.8 Watson’s Expansions
§20.8 Watson’s Expansions
… ►This reference and Bellman (1961, pp. 46–47) include other expansions of this type.5: 4.47 Approximations
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§4.47(i) Chebyshev-Series Expansions
…6: 12.18 Methods of Computation
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►These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions.
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7: 29.16 Asymptotic Expansions
§29.16 Asymptotic Expansions
…8: 6.20 Approximations
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§6.20(ii) Expansions in Chebyshev Series
… ►Luke and Wimp (1963) covers for (20D), and and for (20D).
Luke (1969b, pp. 41–42) gives Chebyshev expansions of , , and for , . The coefficients are given in terms of series of Bessel functions.
Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric -function (§13.2(i)) from which Chebyshev expansions near infinity for , , and follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the functions. If the scheme can be used in backward direction.