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1: 12.10 Uniform Asymptotic Expansions for Large Parameter

§12.10 Uniform Asymptotic Expansions for Large Parameter

… ►In this section we give asymptotic expansions of PCFs for large values of the parameter $a$ that are uniform with respect to the variable $z$, when both $a$ and $z$ $(=x)$ are real. … ►The modified expansion (12.10.31) shares the property of (12.10.3) that it applies when $\mu \to \mathrm{\infty }$ uniformly with respect to $t\in [1+\delta ,\mathrm{\infty })$. … … ►

Modified Expansions

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

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§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

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

Referenced by:

§5.11(ii), §5.11(ii).

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3: Bibliography W

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Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules. Ann. Inst. Statist. Math. 45 (3), pp. 467–475.

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ISSN 0020-3157, Document, MathReview (N. K. Thakare), ZentralBlatt

Referenced by:

§35.9.

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On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.

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Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§8.11(v).

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Asymptotic expansions of the Kontorovich-Lebedev transform. Appl. Anal. 12 (3), pp. 161–172.

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ISSN 0003-6811, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§10.43(v).

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Applications of some recent results in asymptotic expansions. Congr. Numer. 37, pp. 145–182.

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ISSN 0384-9864, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§19.27(vi).

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The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc. (2) 38, pp. 257–270.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.46.

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4: 2.11 Remainder Terms; Stokes Phenomenon

… ►The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. ►A simple example is provided by Euler’s transformation (§3.9(ii)) applied to the asymptotic expansion for the exponential integral (§6.12(i)): … ► … ►shows that this direct estimate is correct to almost 3D. … ►For example, extrapolated values may converge to an accurate value on one side of a Stokes line (§2.11(iv)), and converge to a quite inaccurate value on the other.

5: Bibliography M

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Asymptotic expansions for the zeros of certain special functions. J. Comput. Appl. Math. 145 (2), pp. 261–267.

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ISSN 0377-0427, Document, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.70, §11.4(vii), §6.13.

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Hill’s Equation. Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney.

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Notes:

Reprinted by Dover Publications, Inc., New York, 1979.

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MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§1.3(iii), §28.29(i), §28.29(ii), §28.29(ii), §28.29(iii), §28.29(iii), §28.29(iii), §28.30(i), Ch.28, §29.3(i), §29.9.

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Explicit error terms for asymptotic expansions of Stieltjes transforms. J. Inst. Math. Appl. 22 (2), pp. 129–145.

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ISSN 0020-2932, Document, MathReview (G. M. Habibullah), ZentralBlatt

Referenced by:

§2.6(ii).

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Applying $q$-Laguerre polynomials to the derivation of $q$-deformed energies of oscillator and Coulomb systems. Romanian Reports in Physics 57 (1), pp. 25–34.

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Referenced by:

§17.17.

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On asymptotic expansions of ellipsoidal wave functions. Math. Nachr. 32, pp. 157–172.

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Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.11, §29.7(ii).

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6: Bibliography F

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On the reversion of an asymptotic expansion and the zeros of the Airy functions. SIAM Rev. 41 (4), pp. 762–773.

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ISSN 1095-7200, Document, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§2.2, §2.2, §3.8(v), §9.9(iv), §9.9(iv).

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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II. J. Math. Anal. Appl. 7 (3), pp. 440–451.

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Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. J. Math. Anal. Appl. 6 (3), pp. 394–403.

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External Links:

Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. III. J. Math. Anal. Appl. 12 (3), pp. 593–601.

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Document, MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§16.11(iii).

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On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.

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ISSN 0036-1410, Document, Link, MathReview (Archibald Brown), ZentralBlatt

Referenced by:

§2.3(vi).

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7: 11.6 Asymptotic Expansions

§11.6 Asymptotic Expansions

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§11.6(i) Large $|z|$, Fixed $\nu$

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§11.6(ii) Large $|\nu |$, Fixed $z$

… ►More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)). …

8: 12.11 Zeros

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§12.11(ii) Asymptotic Expansions of Large Zeros

… ►When $a>-\frac{1}{2}$ the zeros are asymptotically given by $z_{a,s}$ and $\overline{z_{a,s}}$, where $s$ is a large positive integer and … ►

§12.11(iii) Asymptotic Expansions for Large Parameter

►For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …as $\mu$ ($=\sqrt{-2a}$) $\to \mathrm{\infty }$, $s$ fixed. …

9: Bibliography

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Asymptotic expansions of spheroidal wave functions. J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.

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External Links:

MathReview (J. G. van der Corput), ZentralBlatt

Referenced by:

§30.9(iii).

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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.

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Notes:

Includes algorithms for Lerch’s transcendent. C and Mathematica codes by Aksenov and Jentschura are available.

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Code, Document

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1st item.

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Introduction to Interval Computations. Computer Science and Applied Mathematics, Academic Press Inc., New York.

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Notes:

Translated from the German by Jon Rokne

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ISBN 0-12-049820-0, MathReview (H. Ratschek), ZentralBlatt

Referenced by:

§3.1(ii).

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Uniform asymptotic expansions for exponential integrals $E_n(x)$ and Bickley functions ${\mathrm{Ki}}_n(x)$ . ACM Trans. Math. Software 9 (4), pp. 467–479.

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ISSN 0098-3500, Document, MathReview (Marietta J. Tretter), ZentralBlatt

Referenced by:

§10.43(iii).

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Special value of the hypergeometric function ${}_3{}^{}F_2^{}$ and connection formulae among asymptotic expansions. J. Indian Math. Soc. (N.S.) 51, pp. 161–221.

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ISSN 0019-5839, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§16.4(iii).

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10: Bibliography L

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An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

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A note on the uniform asymptotic expansion of integrals with coalescing endpoint and saddle points. J. Phys. A 19 (3), pp. 329–335.

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ISSN 0305-4470, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.4(vi).

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

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

Referenced by:

§19.27(vi).

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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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Document, MathReview (Colin Clark), ZentralBlatt

Referenced by:

§36.12(iii), §36.14(i).

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