About the Project
NIST

Watson expansions

AdvancedHelp

(0.001 seconds)

1—10 of 40 matching pages

1: 20.8 Watson’s Expansions
§20.8 Watson’s Expansions
2: 2.4 Contour Integrals
§2.4(i) Watson’s Lemma
3: 2.3 Integrals of a Real Variable
§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 | z | , Fixed ν
where … and … See also Watson (1944, §10.15). …
5: 11.2 Definitions
§11.2(i) Power-Series Expansions
The expansions (11.2.1) and (11.2.2) are absolutely convergent for all finite values of z . …
6: 11.9 Lommel Functions
and … For the foregoing results and further information see Watson (1944, §§10.7–10.73) and Babister (1967, §3.16).
§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
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)). …
8: 11.10 Anger–Weber Functions
§11.10(iii) Maclaurin Series
These expansions converge absolutely for all finite values of z . …
11.10.19 J - 1 2 ( z ) = E 1 2 ( z ) = ( 1 2 π z ) - 1 2 ( A + ( χ ) cos z - A - ( χ ) sin z ) ,
where …
§11.10(viii) Expansions in Series of Products of Bessel Functions
9: 10.40 Asymptotic Expansions for Large Argument
§10.40 Asymptotic Expansions for Large Argument
§10.40(i) Hankel’s Expansions
Products
§10.40(iv) Exponentially-Improved Expansions
10: 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).