Watson%20expansions
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1: 5.11 Asymptotic Expansions
§5.11 Asymptotic Expansions
… ►Wrench (1968) gives exact values of up to . … ►The expansion (5.11.1) is called Stirling’s series (Whittaker and Watson (1927, §12.33)), whereas the expansion (5.11.3), or sometimes just its leading term, is known as Stirling’s formula (Abramowitz and Stegun (1964, §6.1), Olver (1997b, p. 88)). … ►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). … ►2: 11.6 Asymptotic Expansions
§11.6 Asymptotic Expansions
… ►For re-expansions of the remainder terms in (11.6.1) and (11.6.2), see Dingle (1973, p. 445). … ► … ►See also Watson (1944, p. 336). … ►and for an estimate of the relative error in this approximation see Watson (1944, p. 336).3: Bibliography W
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The cubic transformation of the hypergeometric function.
Quart. J. Pure and Applied Math. 41, pp. 70–79.
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Generating functions of class-numbers.
Compositio Math. 1, pp. 39–68.
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The surface of an ellipsoid.
Quart. J. Math., Oxford Ser. 6, pp. 280–287.
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Two tables of partitions.
Proc. London Math. Soc. (2) 42, pp. 550–556.
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A table of Ramanujan’s function
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Proc. London Math. Soc. (2) 51, pp. 1–13.
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4: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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Asymptotic expansions of spheroidal wave functions.
J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.
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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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Chapter of Ramanujan’s second notebook: Theta-functions and -series.
Mem. Amer. Math. Soc. 53 (315), pp. v+85.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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5: 12.11 Zeros
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§12.11(ii) Asymptotic Expansions of Large Zeros
… ►§12.11(iii) Asymptotic Expansions for Large Parameter
►For large negative values of the real zeros of , , , and can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). … ►
12.11.4
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12.11.9
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