Watson%20lemma
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1: 20 Theta Functions
Chapter 20 Theta Functions
…2: 23 Weierstrass Elliptic and Modular
Functions
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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: 20.8 Watson’s Expansions
5: 11.6 Asymptotic Expansions
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►See also Watson (1944, p. 336).
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►and for an estimate of the relative error in this approximation see Watson (1944, p. 336).
6: 2.3 Integrals of a Real Variable
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§2.3(ii) Watson’s Lemma
… ►(In other words, differentiation of (2.3.8) with respect to the parameter (or ) is legitimate.) … ►Watson’s lemma can be regarded as a special case of this result. ►For error bounds for Watson’s lemma and Laplace’s method see Boyd (1993) and Olver (1997b, Chapter 3). … ►The first result is the analog of Watson’s lemma (§2.3(ii)). …7: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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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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A trinomial analogue of Bailey’s lemma and superconformal invariance.
Comm. Math. Phys. 192 (2), pp. 245–260.
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Bailey’s Transform, Lemma, Chains and Tree.
In Special Functions 2000: Current Perspective and Future
Directions (Tempe, AZ), J. Bustoz, M. E. H. Ismail, and S. K. Suslov (Eds.),
NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, pp. 1–22.
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8: 2.4 Contour Integrals
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§2.4(i) Watson’s Lemma
… ►If this integral converges uniformly at each limit for all sufficiently large , then by the Riemann–Lebesgue lemma (§1.8(i)) …9: 29 Lamé Functions
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