# Hankel expansions

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## 1—10 of 34 matching pages

##### 1: 10.17 Asymptotic Expansions for Large Argument
###### §10.17(v) Exponentially-Improved Expansions
For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).
##### 3: 10.20 Uniform Asymptotic Expansions for Large Order
###### §10.20 Uniform Asymptotic Expansions for Large Order
10.20.6 $\rselection{{H^{(1)}_{\nu}}\left(\nu z\right)\\ {H^{(2)}_{\nu}}\left(\nu z\right)}\sim 2e^{\mp\pi i/3}\left(\frac{4\zeta}{1-z^% {2}}\right)^{\frac{1}{4}}\left(\frac{\operatorname{Ai}\left(e^{\pm 2\pi i/3}% \nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{1}{3}}}\sum_{k=0}^{\infty}\frac{A_{k% }(\zeta)}{\nu^{2k}}+\frac{e^{\pm 2\pi i/3}\operatorname{Ai}'\left(e^{\pm 2\pi i% /3}\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}}}\sum_{k=0}^{\infty}\frac{B% _{k}(\zeta)}{\nu^{2k}}\right),$
10.20.9 $\rselection{{H^{(1)}_{\nu}}'\left(\nu z\right)\\ {H^{(2)}_{\nu}}'\left(\nu z\right)}\sim\frac{4e^{\mp 2\pi i/3}}{z}\left(\frac{% 1-z^{2}}{4\zeta}\right)^{\frac{1}{4}}\*\left(\frac{e^{\mp 2\pi i/3}% \operatorname{Ai}\left(e^{\pm 2\pi i/3}\nu^{\frac{2}{3}}\zeta\right)}{\nu^{% \frac{4}{3}}}\sum_{k=0}^{\infty}\frac{C_{k}(\zeta)}{\nu^{2k}}+\frac{% \operatorname{Ai}'\left(e^{\pm 2\pi i/3}\nu^{\frac{2}{3}}\zeta\right)}{\nu^{% \frac{2}{3}}}\sum_{k=0}^{\infty}\frac{D_{k}(\zeta)}{\nu^{2k}}\right),$
###### §10.20(iii) Double Asymptotic Properties
For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of $z$ see §10.41(v).
##### 4: 10.19 Asymptotic Expansions for Large Order
###### §10.19(iii) Transition Region
10.19.13 $\rselection{{H^{(1)}_{\nu}}'\left(\nu+a\nu^{\frac{1}{3}}\right)\\ {H^{(2)}_{\nu}}'\left(\nu+a\nu^{\frac{1}{3}}\right)}\sim-\frac{2^{\frac{5}{3}}% }{\nu^{\frac{2}{3}}}e^{\pm\pi i/3}\operatorname{Ai}'\left(e^{\mp\pi i/3}2^{% \frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{R_{k}(a)}{\nu^{2k/3}}+\frac{2^{% \frac{4}{3}}}{\nu^{\frac{4}{3}}}e^{\mp\pi i/3}\operatorname{Ai}\left(e^{\mp\pi i% /3}2^{\frac{1}{3}}a\right)\sum_{k=0}^{\infty}\frac{S_{k}(a)}{\nu^{2k/3}},$
##### 5: Bibliography W
• R. Wong (1976) Error bounds for asymptotic expansions of Hankel transforms. SIAM J. Math. Anal. 7 (6), pp. 799–808.
• R. Wong (1977) Asymptotic expansions of Hankel transforms of functions with logarithmic singularities. Comput. Math. Appl. 3 (4), pp. 271–286.
• ##### 6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
$\int_{0}^{1}\left(1+y^{\nu+\frac{1}{2}}\right)\left|f(y)\right|\,\mathrm{d}y<\infty.$
##### 7: 10.41 Asymptotic Expansions for Large Order
###### §10.41(v) Double Asymptotic Properties (Continued)
We first prove that for the expansions (10.20.6) for the Hankel functions ${H^{(1)}_{\nu}}\left(\nu z\right)$ and ${H^{(2)}_{\nu}}\left(\nu z\right)$ the $z$-asymptotic property applies when $z\to\pm i\infty$, respectively. …
##### 8: Bibliography N
• G. Nemes (2017b) Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions. Acta Appl. Math. 150, pp. 141–177.
• ##### 9: 10.1 Special Notation
For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
##### 10: Bibliography G
• E. A. Galapon and K. M. L. Martinez (2014) Exactification of the Poincaré asymptotic expansion of the Hankel integral: spectacularly accurate asymptotic expansions and non-asymptotic scales. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2162), pp. 20130529, 16.