modified expansions in terms of elementary functions

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1: 12.10 Uniform Asymptotic Expansions for Large Parameter

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§12.10(vi) Modifications of Expansions in Elementary Functions

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2: 28.8 Asymptotic Expansions for Large $q$

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§28.8(ii) Sips’ Expansions

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Barrett’s Expansions

… ►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. … ►With additional restrictions on $z$, uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii). ►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions ${\mathrm{me}}_{\nu }(z,q)$ (§28.12(ii)) and modified Mathieu functions ${\mathrm{M}}_{\nu }^{(j)}(z,h)$ (§28.20(iii)). …

3: 18.15 Asymptotic Approximations

… ►These expansions are in terms of Whittaker functions (§13.14). … ►These expansions are in terms of Bessel functions and modified Bessel functions, respectively. … ►

In Terms of Elementary Functions

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In Terms of Bessel Functions

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In Terms of Airy Functions

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4: 2.8 Differential Equations with a Parameter

… ►These are elementary functions in Case I, and Airy functions (§9.2) in Case II. … ►Corresponding to each positive integer $n$ there are solutions $W_{n,j}(u,\xi )$, $j=1,2$, that depend on arbitrarily chosen reference points ${\alpha }_j$, are $C^{\mathrm{\infty }}$ or analytic on $𝚫$, and as $u\to \mathrm{\infty }$ … ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Airy functions see especially §10.20 and §§12.10(vii), 12.10(viii); also §§12.14(ix), 13.20(v), 13.21(iii), 13.21(iv), 15.12(iii), 18.15(iv), 30.9(i), 30.9(ii), 32.11(ii), 32.11(iii), 33.12(i), 33.12(ii), 33.20(iv), 36.12(ii), 36.13. … ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). … ►For further examples of uniform asymptotic approximations in terms of Bessel functions or modified Bessel functions of variable order see §§13.21(ii), 14.15(ii), 14.15(iv), 14.20(viii), 30.9(i), 30.9(ii). …

5: 33.20 Expansions for Small $|ϵ|$

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§33.20(i) Case $ϵ=0$

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§33.20(ii) Power-Series in $ϵ$ for the Regular Solution

… ►The functions $Y$ and $K$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by (33.20.6). ►

§33.20(iv) Uniform Asymptotic Expansions

… ►These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$.

6: 14.15 Uniform Asymptotic Approximations

… ►In other words, the convergent hypergeometric series expansions of ${𝖯}_{\nu }^{-\mu }\left(\pm x\right)$ are also generalized (and uniform) asymptotic expansions as $\mu \to \mathrm{\infty }$, with scale $1/\mathrm{\Gamma }\left(j+1+\mu \right)$, $j=0,1,2,\mathrm{\dots }$; compare §2.1(v). … ►Here $I$ and $K$ are the modified Bessel functions (§10.25(ii)). … ►For asymptotic expansions and explicit error bounds, see Dunster (2003b). … ►For convergent series expansions see Dunster (2004). … ►See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials $P_n\left(\cos \theta \right)$ as $n\to \mathrm{\infty }$ with $\theta$ fixed. …

7: 18.32 OP’s with Respect to Freud Weights

… ►A Freud weight is a weight function of the form …These conditions on $Q(x)$ have been strengthened and also relaxed in literature. …However, for asymptotic approximations in terms of elementary functions for the OP’s, and also for their largest zeros, see Levin and Lubinsky (2001) and Nevai (1986). For a uniform asymptotic expansion in terms of Airy functions (§9.2) for the OP’s in the case $Q(x)=x^4$ see Bo and Wong (1999). … ►All of these forms appear in applications, see §18.39(iii) and Table 18.39.1, albeit sometimes with $x\in [0,\mathrm{\infty })$, where the term half-Freud weight is used; or on $x\in [-1,1]$ or $[0,1]$, where the term Rys weight is employed, see Rys et al. (1983). …

8: 3.10 Continued Fractions

… ►if the expansion of its $n$th convergent $C_n$ in ascending powers of $z$ agrees with (3.10.7) up to and including the term in $z^{n-1}$, $n=1,2,3,\mathrm{\dots }$. … ►We say that it is associated with the formal power series $f(z)$ in (3.10.7) if the expansion of its $n$th convergent $C_n$ in ascending powers of $z$, agrees with (3.10.7) up to and including the term in $z^{2n-1}$, $n=1,2,3,\mathrm{\dots }$. … ►For elementary functions, see §§ 4.9 and 4.35. … ►The $A_n$ and $B_n$ of (3.10.2) can be computed by means of three-term recurrence relations (1.12.5). … ►In Gautschi (1979c) the forward series algorithm is used for the evaluation of a continued fraction of an incomplete gamma function (see §8.9). …

9: Bibliography C

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An Elementary Treatise on Elliptic Functions. George Bell and Sons, London.

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Notes:

Republished by Dover Publications, Inc,, New York, 1961. Table erratum: Math. Comp. v. 29 (1975), no. 130, p. 670.

External Links:

ZentralBlatt

Referenced by:

§19.12, Ch.19, A. Cayley (1961).

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BibTeX

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An Elementary Treatise on Elliptic Functions. Dover Publications, New York (English).

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Notes:

Second edition of Cayley (1895), unabridged and corrected

External Links:

ZentralBlatt

Referenced by:

§19.11(i), Erratum (V1.0.24) for Subsection 19.11(i).

Encodings:

BibTeX

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Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.

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External Links:

MathReview (I. I. Hirschman, Jr.), ZentralBlatt

Referenced by:

§12.16.

Encodings:

BibTeX

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Software Manual for the Elementary Functions. Prentice-Hall, Englewood Cliffs.

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External Links:

ISBN 0138220646, ZentralBlatt

Referenced by:

§4.45(i).

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BibTeX

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Algorithm 714: CELEFUNT – A portable test package for complex elementary functions. ACM Trans. Math. Software 19 (1), pp. 1–21.

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External Links:

ISSN 0098-3500, Document, GAMS (module 714; package TOMS), ZentralBlatt

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1st item.

Encodings:

BibTeX

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10: Bibliography L

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Evaluating elementary functions with guaranteed precision. Programming and Computer Software 27 (2), pp. 101–110.

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External Links:

ISSN 0361-7688, Document, MathReview Entry, ZentralBlatt

Referenced by:

§4.48(ii).

Encodings:

BibTeX

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Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions. Stud. Appl. Math. 103 (3), pp. 241–258.

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External Links:

ISSN 0022-2526, Document, Link, MathReview, ZentralBlatt

Referenced by:

§24.11, §24.11.

Encodings:

BibTeX

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Monotonicity in terms of order of the zeros of the derivatives of Bessel functions. Proc. Amer. Math. Soc. 108 (2), pp. 387–389.

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External Links:

ISSN 0002-9939, FullText, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§10.21(iv), §10.21(iv).

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BibTeX

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Integral representation of the Hankel function in terms of parabolic cylinder functions. Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.

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External Links:

Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§12.12, §12.16.

Encodings:

BibTeX

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Highly accurate tables for elementary functions. BIT 35 (3), pp. 352–360.

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External Links:

ISSN 0006-3835, Document, MathReview, ZentralBlatt

Referenced by:

§4.45(i), §4.46.

Encodings:

BibTeX

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