# approximations (except asymptotic)

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## 1—10 of 19 matching pages

##### 1: 19.38 Approximations

###### §19.38 Approximations

…##### 2: 18.15 Asymptotic Approximations

###### §18.15 Asymptotic Approximations

►###### §18.15(i) Jacobi

… ►###### §18.15(ii) Ultraspherical

… ►###### §18.15(iii) Legendre

… ►###### §18.15(iv) Laguerre

…##### 3: 19.27 Asymptotic Approximations and Expansions

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►Although they are obtained (with some exceptions) by approximating uniformly the integrand of each elliptic integral, some occur also as the leading terms of known asymptotic series with error bounds (Wong (1983, §4), Carlson and Gustafson (1985), López (2000, 2001)).
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##### 4: 2.4 Contour Integrals

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###### §2.4(i) Watson’s Lemma

… ►For examples see Olver (1997b, pp. 315–320). ►###### §2.4(iii) Laplace’s Method

… ►###### §2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method

… ►###### §2.4(vi) Other Coalescing Critical Points

…##### 5: 10.72 Mathematical Applications

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►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.
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►If $f(z)$ has a double zero ${z}_{0}$, or more generally ${z}_{0}$ is a zero of order $m$, $m=2,3,4,\mathrm{\dots}$, then uniform asymptotic approximations (but

*not*expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order $1/(m+2)$. …The order of the approximating Bessel functions, or modified Bessel functions, is $1/(\lambda +2)$, except in the case when $g(z)$ has a double pole at ${z}_{0}$. … ►Then for large $u$ asymptotic approximations of the solutions $w$ can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on $u$ and $\alpha $). …##### 6: 10.21 Zeros

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###### §10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros

… ►###### §10.21(vii) Asymptotic Expansions for Large Order

… ►The approximations that follow in §10.21(viii) do not suffer from this drawback. ►###### §10.21(viii) Uniform Asymptotic Approximations for Large Order

… ►This information includes asymptotic approximations analogous to those given in §§10.21(vi), 10.21(vii), and 10.21(x). …##### 7: 2.7 Differential Equations

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###### §2.7(iii) Liouville–Green (WKBJ) Approximation

►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: ►###### Liouville–Green Approximation Theorem

… ►By approximating … ►The first of these references includes extensions to complex variables and reversions for zeros. …##### 8: 36.13 Kelvin’s Ship-Wave Pattern

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►Then with $g$ denoting the acceleration due to gravity, the wave height is approximately given by
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►When $\rho >1$, that is, everywhere except close to the ship, the integrand oscillates rapidly.
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►The disturbance $z(\rho ,\varphi )$ can be approximated by the method of uniform asymptotic approximation for the case of two coalescing stationary points (36.12.11), using the fact that ${\theta}_{\pm}(\varphi )$ are real for $$ and complex for $|\varphi |>{\varphi}_{c}$.
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##### 9: 2.6 Distributional Methods

###### §2.6 Distributional Methods

… ►###### §2.6(ii) Stieltjes Transform

… ►Corresponding results for the*generalized Stieltjes transform*…An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). … ►For rigorous derivations of these results and also order estimates for ${\delta}_{n}(x)$, see Wong (1979) and Wong (1989, Chapter 6).

##### 10: Bibliography K

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Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity.
J. Comput. Appl. Math. 205 (1), pp. 186–206.
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Uniform asymptotic approximations for the Meixner-Sobolev polynomials.
Anal. Appl. (Singap.) 10 (3), pp. 345–361.
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Exponentially accurate uniform asymptotic approximations for integrals and Bleistein’s method revisited.
Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2153), pp. 20130008, 12.
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Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters.
J. B. Wolters, Groningen.
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Zeros of exceptional Hermite polynomials.
J. Approx. Theory 200, pp. 28–39.
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