comparison with Clenshaw–Curtis formula

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11: Bibliography T

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On the connection formula for the first Painlevé equation—from the viewpoint of the exact WKB analysis. Sūrikaisekikenkyūsho Kōkyūroku (931), pp. 70–99.

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Notes:

Painlevé functions and asymptotic analysis (Japanese) (Kyoto, 1995)

External Links:

FullText, MathReview (Andrei A. Kapaev)

Referenced by:

§32.12(i).

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Explicit formulas for the Bernoulli and Euler polynomials and numbers. Abh. Math. Sem. Univ. Hamburg 61, pp. 175–180.

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External Links:

ISSN 0025-5858, Document, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.4(v), §24.6.

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Is Gauss quadrature better than Clenshaw-Curtis?. SIAM Rev. 50 (1), pp. 67–87.

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External Links:

ISSN 0036-1445, Document, MathReview (Kai Diethelm), ZentralBlatt

Referenced by:

§3.5(iv).

Encodings:

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12: Bibliography E

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The Euler-Maclaurin formula revisited. J. Austral. Math. Soc. Ser. B 40 (E), pp. E27–E76 (electronic).

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External Links:

ISSN 0334-2700, FullText, MathReview (J. Németh), ZentralBlatt

Referenced by:

§2.10(i).

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A formula including Legendre’s $E{K}^{\prime }+K{E}^{\prime }-K{K}^{\prime }=\frac{1}{2}\pi$ . Messenger of Math. 33, pp. 31–32.

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Notes:

Errata on p. 45 of the same volume.

External Links:

ZentralBlatt

Referenced by:

§15.16.

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A comparison of some methods for the evaluation of highly oscillatory integrals. J. Comput. Appl. Math. 112 (1-2), pp. 55–69.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§3.5(vii).

Encodings:

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13: DLMF Project News

error generating summary

14: 27.12 Asymptotic Formulas: Primes

§27.12 Asymptotic Formulas: Primes

… ►Descriptions and comparisons of pseudoprime tests are given in Bressoud and Wagon (2000, §§2.4, 4.2, and 8.2) and Crandall and Pomerance (2005, §§3.4–3.6). …

15: 20.9 Relations to Other Functions

… ►The relations (20.9.1) and (20.9.2) between $k$ and $\tau$ (or $q$) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). …

16: 28.34 Methods of Computation

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Representations for $w_{\text{I}}(\pi ;a,\pm q)$ with limit formulas for special solutions of the recurrence relations §28.4(ii) for fixed $a$ and $q$; see Schäfke (1961a).

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Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

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17: 36.15 Methods of Computation

… ►Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. … ►For details, see Connor and Curtis (1982) and Kirk et al. (2000). …

18: Bibliography F

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Algorithms and codes for the Macdonald function: Recent progress and comparisons. J. Comput. Appl. Math. 161 (1), pp. 179–192.

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External Links:

ISSN 0377-0427, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.77(vi).

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The Edmonds asymptotic formulas for the $3j$ and $6j$ symbols. J. Math. Phys. 39 (7), pp. 3906–3915.

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ISSN 0022-2488, Document, MathReview (A. J. Coleman), ZentralBlatt

Referenced by:

§34.8.

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On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.

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External Links:

MathReview (A. G. Law), ZentralBlatt

Referenced by:

§18.32, §32.15.

Encodings:

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19: Bibliography L

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Numerical evaluation of integrals containing a spherical Bessel function by product integration. J. Math. Phys. 22 (7), pp. 1399–1413.

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External Links:

ISSN 0022-2488, Document, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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Comparison of a pair of upper bounds for a ratio of gamma functions. Math. Balkanica (N.S.) 16 (1-4), pp. 195–202.

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External Links:

ISSN 0205-3217, MathReview, ZentralBlatt

Referenced by:

§5.6(i).

Encodings:

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20: 15.2 Definitions and Analytical Properties

… ►Because of the analytic properties with respect to $a$, $b$, and $c$, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. … ►This formula is also valid when $c=-m-\mathrm{\ell }$, $\mathrm{\ell }=0,1,2,\mathrm{\dots }$, provided that we use the interpretation … ►For comparison of $F(a,b;c;z)$ and $𝐅(a,b;c;z)$, with the former using the limit interpretation (15.2.5), see Figures 15.3.6 and 15.3.7. … ►Formula (15.4.6) reads $F(a,b;a;z)={(1-z)}^{-b}$. …