# application to asymptotic expansions

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## 11—20 of 52 matching pages

##### 11: 2.2 Transcendental Equations

###### §2.2 Transcendental Equations

… ►An important case is the reversion of asymptotic expansions for zeros of special functions. …where ${F}_{0}={f}_{0}$ and $s{F}_{s}$ ($s\ge 1$) is the coefficient of ${x}^{-1}$ in the asymptotic expansion of ${(f(x))}^{s}$ (*Lagrange’s formula for the reversion of series*). …Applications to real and complex zeros of Airy functions are given in Fabijonas and Olver (1999). For other examples see de Bruijn (1961, Chapter 2).

##### 12: 12.16 Mathematical Applications

###### §12.16 Mathematical Applications

… ►In Brazel et al. (1992) exponential asymptotics are considered in connection with an eigenvalue problem involving PCFs. … ►Integral transforms and sampling expansions are considered in Jerri (1982).##### 13: 29.7 Asymptotic Expansions

###### §29.7 Asymptotic Expansions

►###### §29.7(i) Eigenvalues

… ►###### §29.7(ii) Lamé Functions

… ►In Müller (1966c) it is shown how these expansions lead to asymptotic expansions for the Lamé functions ${\mathit{Ec}}_{\nu}^{m}(z,{k}^{2})$ and ${\mathit{Es}}_{\nu}^{m}(z,{k}^{2})$. Weinstein and Keller (1985) give asymptotics for solutions of Hill’s equation (§28.29(i)) that are applicable to the Lamé equation.##### 14: Bibliography D

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Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory.
Comm. Pure Appl. Math. 52 (11), pp. 1335–1425.
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Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions.
Proc. Roy. Soc. London Ser. A 440, pp. 37–54.
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Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function.
Proc. Roy. Soc. London Ser. A 452, pp. 1331–1349.
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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions.
Stud. Appl. Math. 107 (3), pp. 293–323.
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Convergent expansions for solutions of linear ordinary differential equations having a simple pole, with an application to associated Legendre functions.
Stud. Appl. Math. 113 (3), pp. 245–270.
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##### 15: 2.9 Difference Equations

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►For asymptotic expansions in inverse factorial series see Olde Daalhuis (2004a).
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►Error bounds and applications are included.
►For discussions of turning points, transition points, and uniform asymptotic expansions for solutions of linear difference equations of the second order see Wang and Wong (2003, 2005).
►For an introduction to, and references for, the general asymptotic theory of linear difference equations of arbitrary order, see Wimp (1984, Appendix B).
►For applications of asymptotic methods for difference equations to orthogonal polynomials, see, e.
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##### 16: 6.18 Methods of Computation

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►For large $x$ and $\left|z\right|$, expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available.
The attainable accuracy of the asymptotic expansions can be increased considerably by exponential improvement.
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►For an application of the Gauss–Legendre formula (§3.5(v)) see Tooper and Mark (1968).
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►Power series, asymptotic expansions, and quadrature can also be used to compute the functions $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$.
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►Zeros of $\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ can be computed to high precision by Newton’s rule (§3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations.
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##### 17: 33.12 Asymptotic Expansions for Large $\eta $

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►Then, by application of the results given in §§2.8(iii) and 2.8(iv), two sets of asymptotic expansions can be constructed for ${F}_{\mathrm{\ell}}(\eta ,\rho )$ and ${G}_{\mathrm{\ell}}(\eta ,\rho )$ when $\eta \to \mathrm{\infty}$.
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##### 18: Bibliography O

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Uniform asymptotic expansions for hypergeometric functions with large parameters. I.
Analysis and Applications (Singapore) 1 (1), pp. 111–120.
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Uniform asymptotic expansions for hypergeometric functions with large parameters. II.
Analysis and Applications (Singapore) 1 (1), pp. 121–128.
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Uniform asymptotic expansions for hypergeometric functions with large parameters. III.
Analysis and Applications (Singapore) 8 (2), pp. 199–210.
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On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions.
J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 225–243.
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Applications of Lie Groups to Differential Equations.
2nd edition, Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York.
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##### 19: 10.72 Mathematical Applications

###### §10.72 Mathematical Applications

►###### §10.72(i) Differential Equations with Turning Points

►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … ►These asymptotic expansions are uniform with respect to $z$, including cut neighborhoods of ${z}_{0}$, and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation. … ►##### 20: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function

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►For asymptotic expansions of $\varphi (\rho ,\beta ;z)$ as $z\to \mathrm{\infty}$ in various sectors of the complex $z$-plane for fixed real values of $\rho $ and fixed real or complex values of $\beta $, see Wright (1935) when $\rho >0$, and Wright (1940b) when $$.
For exponentially-improved asymptotic expansions in the same circumstances, together with smooth interpretations of the corresponding Stokes phenomenon (§§2.11(iii)–2.11(v)) see Wong and Zhao (1999b) when $\rho >0$, and Wong and Zhao (1999a) when $$.
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►This reference includes exponentially-improved asymptotic expansions for ${E}_{a,b}\left(z\right)$ when $|z|\to \mathrm{\infty}$, together with a smooth interpretation of Stokes phenomena.
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►For incomplete modified Bessel functions and Hankel functions, including applications, see Cicchetti and Faraone (2004).
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