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1: 13.19 Asymptotic Expansions for Large Argument

§13.19 Asymptotic Expansions for Large Argument

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2: 10.40 Asymptotic Expansions for Large Argument

§10.40 Asymptotic Expansions for Large Argument

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Products

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$\nu$-Derivative

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§10.40(iv) Exponentially-Improved Expansions

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3: 10.17 Asymptotic Expansions for Large Argument

§10.17 Asymptotic Expansions for Large Argument

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§10.17(ii) Asymptotic Expansions of Derivatives

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§10.17(iii) Error Bounds for Real Argument and Order

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§10.17(v) Exponentially-Improved Expansions

… ►For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).

4: 13.7 Asymptotic Expansions for Large Argument

§13.7 Asymptotic Expansions for Large Argument

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§13.7(ii) Error Bounds

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§13.7(iii) Exponentially-Improved Expansion

… ►For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).

5: Bibliography Y

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Computation of Kummer functions $U(a,b,x)$ for large argument $x$ by using the $\tau$-method. Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).

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Notes:

Japanese with English summary. Double-precision Fortran.

External Links:

ISSN 0387-5806, FullText, MathReview

Referenced by:

§13.32(ii).

Encodings:

BibTeX

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6: 10.67 Asymptotic Expansions for Large Argument

§10.67 Asymptotic Expansions for Large Argument

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§10.67(i) ${\mathrm{ber}}_{\nu }x,{\mathrm{bei}}_{\nu }x,{\ker }_{\nu }x,{\mathrm{kei}}_{\nu }x$, and Derivatives

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§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu =0$

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7: Bibliography N

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Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes. ACM Trans. Math. Software 18 (3), pp. 345–349.

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Notes:

Double-precision Fortran, minimum accuracy 9S, maximum accuracy 13S.

External Links:

ISSN 0098-3500, Document, GAMS (module 707; package TOMS), ZentralBlatt

Referenced by:

1st item.

Encodings:

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Numerical evaluation of the confluent hypergeometric function for complex arguments of large magnitudes. J. Comput. Appl. Math. 39 (2), pp. 193–200.

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Notes:

Extended-precision Fortran evaluator, minimum accuracy 9S, maximum accuracy 13S.

External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§13.32(iii).

Encodings:

BibTeX

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On the large argument asymptotics of the Lommel function via Stieltjes transforms. Asymptot. Anal. 91 (3-4), pp. 265–281.

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External Links:

ISSN 0921-7134, Document, MathReview Entry, ZentralBlatt

Referenced by:

§11.6(iii), §11.6(iii), §11.9(iii), §11.9(iii).

Encodings:

BibTeX

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Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions. Acta Appl. Math. 150, pp. 141–177.

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External Links:

ISSN 0167-8019, Document, MathReview Entry, ZentralBlatt

Referenced by:

(9.7.16), (9.7.17), §9.7(iii), §9.7(iv).

Encodings:

BibTeX

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Error bounds for the large-argument asymptotic expansions of the Lommel and allied functions. Stud. Appl. Math. 140 (4), pp. 508–541.

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External Links:

ISSN 0022-2526, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt

Referenced by:

§11.11(i), §11.11(i).

Encodings:

BibTeX

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8: 11.13 Methods of Computation

… ►Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. …

9: 10.18 Modulus and Phase Functions

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§10.18(iii) Asymptotic Expansions for Large Argument

… ►In (10.18.17) and (10.18.18) the remainder after $n$ terms does not exceed the $(n+1)$th term in absolute value and is of the same sign, provided that $n>\nu -\frac{1}{2}$ for (10.18.17) and $-\frac{3}{2}\le \nu \le \frac{3}{2}$ for (10.18.18).

10: 10.68 Modulus and Phase Functions

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§10.68(iii) Asymptotic Expansions for Large Argument

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