# uniform expansions

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##### 2: 9.15 Mathematical Applications
Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …
##### 3: 33.12 Asymptotic Expansions for Large $\eta$
###### §33.12(ii) UniformExpansions
The first set is in terms of Airy functions and the expansions are uniform for fixed $\ell$ and $\delta\leq z<\infty$, where $\delta$ is an arbitrary small positive constant. …
##### 4: 18.40 Methods of Computation
Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree. …
##### 6: 10.57 Uniform Asymptotic Expansions for Large Order
###### §10.57 Uniform Asymptotic Expansions for Large Order
Asymptotic expansions for $\mathsf{j}_{n}\left((n+\tfrac{1}{2})z\right)$, $\mathsf{y}_{n}\left((n+\tfrac{1}{2})z\right)$, ${\mathsf{h}^{(1)}_{n}}\left((n+\tfrac{1}{2})z\right)$, ${\mathsf{h}^{(2)}_{n}}\left((n+\tfrac{1}{2})z\right)$, ${\mathsf{i}^{(1)}_{n}}\left((n+\tfrac{1}{2})z\right)$, and $\mathsf{k}_{n}\left((n+\tfrac{1}{2})z\right)$ as $n\to\infty$ that are uniform with respect to $z$ can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). …
##### 7: 8.25 Methods of Computation
DiDonato and Morris (1986) describes an algorithm for computing $P\left(a,x\right)$ and $Q\left(a,x\right)$ for $a\geq 0$, $x\geq 0$, and $a+x\neq 0$ from the uniform expansions in §8.12. …
##### 8: Bibliography Q
• W.-Y. Qiu and R. Wong (2000) Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.
• ##### 9: 10.41 Asymptotic Expansions for Large Order
###### §10.41(ii) UniformExpansions for Real Variable
10.41.4 $K_{\nu}\left(\nu z\right)\sim\left(\frac{\pi}{2\nu}\right)^{\frac{1}{2}}\frac{% e^{-\nu\eta}}{(1+z^{2})^{\frac{1}{4}}}\sum_{k=0}^{\infty}(-1)^{k}\frac{U_{k}(p% )}{\nu^{k}},$
###### §10.41(iii) UniformExpansions for Complex Variable
Similar analysis can be developed for the uniform asymptotic expansions in terms of Airy functions given in §10.20. …
##### 10: 8.12 Uniform Asymptotic Expansions for Large Parameter
###### §8.12 Uniform Asymptotic Expansions for Large Parameter
A different type of uniform expansion with coefficients that do not possess a removable singularity at $z=a$ is given by …
###### Inverse Function
These expansions involve the inverse error function $\operatorname{inverfc}\left(x\right)$7.17), and are uniform with respect to $q\in[0,1]$. …