application to asymptotic expansions

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21: Bibliography L

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Uniform asymptotic expansions of symmetric elliptic integrals. Constr. Approx. 17 (4), pp. 535–559.

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External Links:

ISSN 0176-4276, Document, MathReview (Valdir A. Menegatto), ZentralBlatt

Referenced by:

§19.27(vi).

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A systematic “saddle point near a pole” asymptotic method with application to the Gauss hypergeometric function. Stud. Appl. Math. 127 (1), pp. 24–37.

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ISSN 0022-2526, Document, FullText, MathReview (M. S. Mahadeva Naika), ZentralBlatt

Referenced by:

§15.12(ii), §15.12(ii).

Encodings:

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Asymptotic expansions of the Whittaker functions for large order parameter. Methods Appl. Anal. 6 (2), pp. 249–256.

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Notes:

Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part II

External Links:

ISSN 1073-2772, Pdf, MathReview, ZentralBlatt

Referenced by:

§13.21(i), §13.24(ii).

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Uniform asymptotic expansions at a caustic. Comm. Pure Appl. Math. 19, pp. 215–250.

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External Links:

Document, MathReview (Colin Clark), ZentralBlatt

Referenced by:

§36.12(iii), §36.14(i).

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Adjusted forms of the Fourier coefficient asymptotic expansion and applications in numerical quadrature. Math. Comp. 25 (113), pp. 87–104.

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External Links:

FullText, MathReview (J. Bryant), ZentralBlatt

Referenced by:

§2.3(iv).

Encodings:

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22: 14.32 Methods of Computation

… ►Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter. In particular, for small or moderate values of the parameters $\mu$ and $\nu$ the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. … ►

Application of the uniform asymptotic expansions for large values of the parameters given in §§14.15 and 14.20(vii)–14.20(ix).

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23: 28.34 Methods of Computation

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Use of asymptotic expansions and approximations for large $q$ (§§28.8(i), 28.16). See also Zhang and Jin (1996, pp. 482–485).

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Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

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Use of asymptotic expansions and approximations for large $q$ (§§28.8(ii)–28.8(iv)).

►Also, once the eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$, and ${\lambda }_{\nu }\left(q\right)$ have been computed the following methods are applicable: … ►

Use of asymptotic expansions for large $z$ or large $q$. See §§28.25 and 28.26.

24: 10.24 Functions of Imaginary Order

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10.24.1 $$x^2\frac{d^{2}w}{{dx}^2}+x\frac{dw}{dx}+(x^2+{\nu }^2)w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $x$: real variable and $\nu$: complex parameter

Referenced by:

§10.24, §10.24

Permalink:

http://dlmf.nist.gov/10.24.E1

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See also:

Annotations for §10.24 and Ch.10

… ►As $x\to +\mathrm{\infty }$, with $\nu$ fixed, … ►As $x\to 0+$, with $\nu$ fixed, … ►For mathematical properties and applications of ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$, including zeros and uniform asymptotic expansions for large $\nu$, see Dunster (1990a). …

25: Bibliography S

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Uniform asymptotic expansions of modified Mathieu functions. J. Reine Angew. Math. 247, pp. 1–17.

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External Links:

MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

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Asymptotic expansions of Mellin transforms and analogues of Watson’s lemma. SIAM J. Math. Anal. 16 (4), pp. 896–906.

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ISSN 0036-1410, Document, Link, MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.3(vi).

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A simple approach to asymptotic expansions for Fourier integrals of singular functions. Appl. Math. Comput. 216 (11), pp. 3378–3385.

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ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§2.3(iv).

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Asymptotic expansion of Mellin transforms in the complex plane. Int. J. Pure Appl. Math. 71 (3), pp. 465–480.

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External Links:

ISSN 1311-8080, MathReview Entry, ZentralBlatt

Referenced by:

§2.3(vi).

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Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.

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External Links:

Abstract

Referenced by:

§18.16(v).

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26: Bibliography N

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On an asymptotic expansion of the Kontorovich-Lebedev transform. Applicable Anal. 39 (4), pp. 249–263.

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External Links:

ISSN 0003-6811, Document, MathReview (Aloknath Chakrabarti), ZentralBlatt

Referenced by:

§10.43(v).

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On an asymptotic expansion of the Kontorovich-Lebedev transform. Methods Appl. Anal. 3 (1), pp. 98–108.

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External Links:

ISSN 1073-2772, FullText, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§10.43(v).

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Error bounds for the asymptotic expansion of the Hurwitz zeta function. Proc. A. 473 (2203), pp. 20170363, 16.

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ISSN 1364-5021, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.15.

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Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.18(ii), §8.18(ii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).

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Error bounds and exponential improvement for Hermite’s asymptotic expansion for the gamma function. Appl. Anal. Discrete Math. 7 (1), pp. 161–179.

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External Links:

ISSN 1452-8630, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§5.11(ii), §5.11(ii).

Encodings:

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27: Bibliography B

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The Bounds for the Error Term of an Asymptotic Approximation of Jacobi Polynomials. In Orthogonal Polynomials and Their Applications (Segovia, 1986), Lecture Notes in Math., Vol. 1329, pp. 203–221.

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External Links:

MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§18.15(i), §18.15(i).

Encodings:

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Application of uniform asymptotics to the second Painlevé transcendent. Arch. Rational Mech. Anal. 143 (3), pp. 241–271.

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External Links:

ISSN 0945-8396, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.11(ii).

Encodings:

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Uniform asymptotic expansion of Charlier polynomials. Methods Appl. Anal. 1 (3), pp. 294–313.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.24.

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Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions. SIAM J. Math. Anal. 17 (2), pp. 422–450.

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External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.15(iv), §14.15(iv), §14.26, §2.8(vi).

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On the asymptotic expansion of some integrals. Arch. Math. (Basel) 42 (3), pp. 253–259.

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External Links:

ISSN 0003-889X, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§2.5(ii).

Encodings:

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28: 7.20 Mathematical Applications

§7.20 Mathematical Applications

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§7.20(i) Asymptotics

►For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951). ►The complementary error function also plays a ubiquitous role in constructing exponentially-improved asymptotic expansions and providing a smooth interpretation of the Stokes phenomenon; see §§2.11(iii) and 2.11(iv). … ►For applications in statistics and probability theory, also for the role of the normal distribution functions (the error functions and probability integrals) in the asymptotics of arbitrary probability density functions, see Johnson et al. (1994, Chapter 13) and Patel and Read (1982, Chapters 2 and 3).

29: 8.22 Mathematical Applications

§8.22 Mathematical Applications

… ►plays a fundamental role in re-expansions of remainder terms in asymptotic expansions, including exponentially-improved expansions and a smooth interpretation of the Stokes phenomenon. … ►

§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function

… ►See Paris and Cang (1997). … ►For further information on ${\zeta }_x(s)$, including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). …

30: 8.11 Asymptotic Approximations and Expansions

§8.11 Asymptotic Approximations and Expansions

… ►where $\delta$ denotes an arbitrary small positive constant. … ►For an exponentially-improved asymptotic expansion (§2.11(iii)) see Olver (1991a). … ►This reference also contains explicit formulas for $b_k(\lambda )$ in terms of Stirling numbers and for the case $\lambda >1$ an asymptotic expansion for $b_k(\lambda )$ as $k\to \mathrm{\infty }$. … ►