Watson expansions

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1: 20.8 Watson’s Expansions

§20.8 Watson’s Expansions

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2: 2.4 Contour Integrals

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§2.4(i) Watson’s Lemma

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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

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§11.11(i) Large $|z|$, Fixed $\nu$

… ►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 $z$. …

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

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§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

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§11.10(iii) Maclaurin Series

… ►These expansions converge absolutely for all finite values of $z$. … ►

11.10.19 $${\mathbf{J}}_{-\frac{1}{2}}\left(z\right)={\mathbf{E}}_{\frac{1}{2}}\left(z\right)={(\frac{1}{2}\pi z)}^{-\frac{1}{2}}(A_{+}(\chi )\cos z-A_{-}(\chi )\sin z),$$

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Symbols:

${\mathbf{J}}_{\nu }\left(z\right)$: Anger function, ${\mathbf{E}}_{\nu }\left(z\right)$: Weber function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable, $A$: expansion function and $\chi$

Referenced by:

§11.10(vi)

Permalink:

http://dlmf.nist.gov/11.10.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §11.10(vi), §11.10 and Ch.11

… ►where … ►

§11.10(viii) Expansions in Series of Products of Bessel Functions

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9: 10.40 Asymptotic Expansions for Large Argument

§10.40 Asymptotic Expansions for Large Argument

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§10.40(i) Hankel’s Expansions

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Products

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§10.40(iv) Exponentially-Improved Expansions

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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).