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41: 14.32 Methods of Computation

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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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42: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

§28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

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43: 8.18 Asymptotic Expansions of $I_x(a,b)$

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§8.18(i) Large Parameters, Fixed $x$

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§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

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Large $a$, Fixed $b$

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Symmetric Case

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General Case

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44: 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).

45: 33.11 Asymptotic Expansions for Large $\rho$

§33.11 Asymptotic Expansions for Large $\rho$

►For large $\rho$, with $\mathrm{\ell }$ and $\eta$ fixed, …

46: 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 $z$

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§11.6(iii) Large $|\nu |$, Fixed $z/\nu$

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47: Bibliography T

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Laguerre polynomials: Asymptotics for large degree. Technical report Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.

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External Links:

FullText

Referenced by:

§2.4(vi).

Encodings:

BibTeX

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On the computation of the incomplete gamma functions for large values of the parameters. In Algorithms for approximation (Shrivenham, 1985), Inst. Math. Appl. Conf. Ser. New Ser., Vol. 10, pp. 479–489.

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External Links:

FullText, MathReview (G. Meinardus), ZentralBlatt

Referenced by:

§8.25(iii).

Encodings:

BibTeX

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Large parameter cases of the Gauss hypergeometric function. J. Comput. Appl. Math. 153 (1-2), pp. 441–462.

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External Links:

ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters. Integral Transforms Spec. Funct. 33 (1), pp. 16–31.

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External Links:

ISSN 1065-2469, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§13.8(iv).

Encodings:

BibTeX

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Evaluation of the exponential integral for large complex arguments. J. Research Nat. Bur. Standards 52, pp. 313–317.

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External Links:

MathReview (H. Bückner), ZentralBlatt

Referenced by:

§6.18(i).

Encodings:

BibTeX

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48: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

§33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

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§33.5(iv) Large $\mathrm{\ell }$

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49: 6.18 Methods of Computation

… ►For large $x$ or $|z|$ these series suffer from slow convergence or cancellation (or both). … ►For large $x$ and $\left|z\right|$, expansions in inverse factorial series (§6.10(i)) or asymptotic expansions (§6.12) are available. … ► $A_0$, $B_0$, and $C_0$ can be computed by Miller’s algorithm (§3.6(iii)), starting with initial values $(A_N,B_N,C_N)=(1,0,0)$, say, where $N$ is an arbitrary large integer, and normalizing via $C_0=1/z$. …

50: 15.19 Methods of Computation

… ►Large values of $|a|$ or $|b|$, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. … ►Initial values for moderate values of $|a|$ and $|b|$ can be obtained by the methods of §15.19(i), and for large values of $|a|$, $|b|$, or $|c|$ via the asymptotic expansions of §§15.12(ii) and 15.12(iii). ►For example, in the half-plane $\mathrm{\Re }z\le \frac{1}{2}$ we can use (15.12.2) or (15.12.3) to compute $F(a,b;c+N+1;z)$ and $F(a,b;c+N;z)$, where $N$ is a large positive integer, and then apply (15.5.18) in the backward direction. …