Watson expansions
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1: 20.8 Watson’s Expansions
§20.8 Watson’s Expansions
…2: 2.4 Contour Integrals
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§2.4(i) Watson’s Lemma
…3: 2.3 Integrals of a Real Variable
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§2.3(ii) Watson’s Lemma
… ►The desired uniform expansion is then obtained formally as in Watson’s lemma and Laplace’s method. …4: 11.11 Asymptotic Expansions of Anger–Weber Functions
§11.11 Asymptotic Expansions of Anger–Weber Functions
►§11.11(i) Large , Fixed
… ►where … ►and … ►See also Watson (1944, §10.15). …5: 11.2 Definitions
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§11.2(i) Power-Series Expansions
… ►The expansions (11.2.1) and (11.2.2) are absolutely convergent for all finite values of . …6: 11.9 Lommel Functions
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►and
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►For the foregoing results and further information see Watson (1944, §§10.7–10.73) and Babister (1967, §3.16).
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§11.9(ii) Expansions in Series of Bessel Functions
… ►§11.9(iii) Asymptotic Expansion
… ►For further information on Lommel functions see Watson (1944, §§10.7–10.75) and Babister (1967, Chapter 3). …7: 5.11 Asymptotic Expansions
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►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)).
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8: 11.10 Anger–Weber Functions
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§11.10(iii) Maclaurin Series
… ►These expansions converge absolutely for all finite values of . … ►
11.10.19
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►where
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