Watson expansions

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1: 20.8 Watson’s Expansions

§20.8 Watson’s Expansions

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2: 11.11 Asymptotic Expansions of Anger–Weber Functions

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11.11.8 $${𝐀}_{\nu }\left(\lambda \nu \right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(2k)!a_k(\lambda )}{{\nu }^{2k+1}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$,

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Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $a_k(\lambda )$: expansion function

Sources:

Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c)

Referenced by:

(11.11.18), (11.11.19), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E8

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which originally was $\nu \to \mathrm{\infty }$, , has been replaced to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.10 $${𝐀}_{-\nu }\left(\lambda \nu \right)\sim -\frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(2k)!a_k(-\lambda )}{{\nu }^{2k+1}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

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Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $a_k(\lambda )$: expansion function

Sources:

Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c)

Proof sketch:

Apply Laplace’s method to the integral (11.10.4).

Referenced by:

(11.11.18), (11.11.19), §11.11(iii), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E10

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which was originally $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.18 $${𝐉}_{\nu }\left(\nu \right)\sim \frac{2^{1/3}}{3^{2/3}\mathrm{\Gamma }\left(\frac{2}{3}\right){\nu }^{1/3}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$,

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\nu$: real or complex order and $\delta$: arbitrary small positive constant

Source:

Olver (1997b, pp. 103 and 352)

Proof sketch:

Dervivable using (11.10.15), (11.10.16), (11.11.8), (11.11.10), (11.11.16) and the expansions in Watson (1944, §8.42).

Referenced by:

§11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E18

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which originally was $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.19 $${𝐄}_{\nu }\left(\nu \right)\sim \frac{2^{1/3}}{3^{7/6}\mathrm{\Gamma }\left(\frac{2}{3}\right){\nu }^{1/3}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐄}_{\nu }\left(z\right)$: Weber function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $\nu$: real or complex order and $\delta$: arbitrary small positive constant

Source:

Olver (1997b, pp. 103 and 352)

Proof sketch:

Dervivable using (11.10.15), (11.10.16), (11.11.8), (11.11.10), (11.11.16) and the expansions in Watson (1944, §8.42).

Referenced by:

§11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E19

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which originally was $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

3: 2.4 Contour Integrals

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§2.4(i) Watson’s Lemma

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4: 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. …

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

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Symbols:

${𝐉}_{\nu }\left(z\right)$: Anger function, ${𝐄}_{\nu }\left(z\right)$: 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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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).