# uniform asymptotic expansions for large order

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##### 3: 10.20 Uniform Asymptotic Expansions for Large Order
###### §10.20 UniformAsymptoticExpansions for LargeOrder
10.20.5 $Y_{\nu}\left(\nu z\right)\sim-\left(\frac{4\zeta}{1-z^{2}}\right)^{\frac{1}{4}% }\left(\frac{\operatorname{Bi}\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{% 1}{3}}}\sum_{k=0}^{\infty}\frac{A_{k}(\zeta)}{\nu^{2k}}+\frac{\operatorname{Bi% }'\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}}}\sum_{k=0}^{\infty}% \frac{B_{k}(\zeta)}{\nu^{2k}}\right),$
10.20.9 $\rselection{{H^{(1)}_{\nu}}'\left(\nu z\right)\\ {H^{(2)}_{\nu}}'\left(\nu z\right)}\sim\frac{4e^{\mp 2\pi i/3}}{z}\left(\frac{% 1-z^{2}}{4\zeta}\right)^{\frac{1}{4}}\*\left(\frac{e^{\mp 2\pi i/3}% \operatorname{Ai}\left(e^{\pm 2\pi i/3}\nu^{\frac{2}{3}}\zeta\right)}{\nu^{% \frac{4}{3}}}\sum_{k=0}^{\infty}\frac{C_{k}(\zeta)}{\nu^{2k}}+\frac{% \operatorname{Ai}'\left(e^{\pm 2\pi i/3}\nu^{\frac{2}{3}}\zeta\right)}{\nu^{% \frac{2}{3}}}\sum_{k=0}^{\infty}\frac{D_{k}(\zeta)}{\nu^{2k}}\right),$
For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of $z$ see §10.41(v).
##### 4: 10.1 Special Notation
For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
##### 5: 10.41 Asymptotic Expansions for Large Order
###### §10.41(ii) UniformExpansions for Real Variable
10.41.4 $K_{\nu}\left(\nu z\right)\sim\left(\frac{\pi}{2\nu}\right)^{\frac{1}{2}}\frac{% e^{-\nu\eta}}{(1+z^{2})^{\frac{1}{4}}}\sum_{k=0}^{\infty}(-1)^{k}\frac{U_{k}(p% )}{\nu^{k}},$
##### 6: 10.24 Functions of Imaginary Order
For mathematical properties and applications of $\widetilde{J}_{\nu}\left(x\right)$ and $\widetilde{Y}_{\nu}\left(x\right)$, including zeros and uniform asymptotic expansions for large $\nu$, see Dunster (1990a). …
##### 7: 10.45 Functions of Imaginary Order
For properties of $\widetilde{I}_{\nu}\left(x\right)$ and $\widetilde{K}_{\nu}\left(x\right)$, including uniform asymptotic expansions for large $\nu$ and zeros, see Dunster (1990a). …
##### 10: Bibliography O
• F. W. J. Olver (1959) Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders. J. Res. Nat. Bur. Standards Sect. B 63B, pp. 131–169.