# Watson expansions

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##### 2: 11.11 Asymptotic Expansions of Anger–Weber Functions
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11.11.8 $\mathbf{A}_{\nu}\left(\lambda\nu\right)\sim\frac{1}{\pi}\sum_{k=0}^{\infty}% \frac{(2k)!\,a_{k}(\lambda)}{\nu^{2k+1}},$ $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta$,
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11.11.10 $\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim-\frac{1}{\pi}\sum_{k=0}^{\infty}% \frac{(2k)!\,a_{k}(-\lambda)}{\nu^{2k+1}},$ $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta$.
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11.11.18 $\mathbf{J}_{\nu}\left(\nu\right)\sim\frac{2^{1/3}}{3^{2/3}\Gamma\left(\tfrac{2% }{3}\right)\nu^{1/3}},$ $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta$,
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11.11.19 $\mathbf{E}_{\nu}\left(\nu\right)\sim\frac{2^{1/3}}{3^{7/6}\Gamma\left(\tfrac{2% }{3}\right)\nu^{1/3}},$ $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta$.
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##### 4: 2.3 Integrals of a Real Variable
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###### §2.3(ii) Watson’s Lemma
βΊThe 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
βΊThe expansions (11.2.1) and (11.2.2) are absolutely convergent for all finite values of $z$. …
##### 6: 11.9 Lommel Functions
βΊand … βΊFor the foregoing results and further information see Watson (1944, §§10.7–10.73) and Babister (1967, §3.16). βΊ
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###### §11.9(iii) Asymptotic Expansion
βΊFor further information on Lommel functions see Watson (1944, §§10.7–10.75) and Babister (1967, Chapter 3). …
##### 7: 5.11 Asymptotic Expansions
βΊThe 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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##### 9: 11.10 Anger–Weber Functions
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###### §11.10(iii) Maclaurin Series
βΊThese expansions converge absolutely for all finite values of $z$. … βΊ
11.10.19 $\mathbf{J}_{-\frac{1}{2}}\left(z\right)=\mathbf{E}_{\frac{1}{2}}\left(z\right)% \\ =(\tfrac{1}{2}\pi z)^{-\frac{1}{2}}(A_{+}(\chi)\cos z-A_{-}(\chi)\sin z),$
βΊwhere … βΊ
##### 10: 11.6 Asymptotic Expansions
###### §11.6 Asymptotic Expansions
βΊFor re-expansions of the remainder terms in (11.6.1) and (11.6.2), see Dingle (1973, p. 445). … βΊβΊSee also Watson (1944, p. 336). … βΊand for an estimate of the relative error in this approximation see Watson (1944, p. 336).