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31: 7.22 Methods of Computation

… ►Additional references are Matta and Reichel (1971) for the application of the trapezoidal rule, for example, to the first of (7.7.2), and Gautschi (1970) and Cuyt et al. (2008) for continued fractions. …

32: 12.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. …

33: 19.13 Integrals of Elliptic Integrals

… ►Cvijović and Klinowski (1994) contains fractional integrals (with free parameters) for $F(\varphi ,k)$ and $E(\varphi ,k)$, together with special cases. …

34: 27.19 Methods of Computation: Factorization

… ►These algorithms include the Continued Fraction Algorithm (cfrac), the Multiple Polynomial Quadratic Sieve (mpqs), the General Number Field Sieve (gnfs), and the Special Number Field Sieve (snfs). …

35: Bibliography J

… ►

Further results on the computation of incomplete gamma functions. In Analytic Theory of Continued Fractions, II (Pitlochry/Aviemore, 1985), W. J. Thron (Ed.), Lecture Notes in Math. 1199, pp. 67–89.

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External Links:

MathReview (Marietta J. Tretter), ZentralBlatt

Referenced by:

§8.25(iv).

Encodings:

BibTeX

… ►

Numerical stability in evaluating continued fractions. Math. Comp. 28 (127), pp. 795–810.

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External Links:

FullText, MathReview (N. N. Abdelmalek), ZentralBlatt

Referenced by:

§3.10(iii).

Encodings:

BibTeX

… ►

Continued Fractions: Analytic Theory and Applications. Encyclopedia of Mathematics and its Applications, Vol. 11, Addison-Wesley Publishing Co., Reading, MA.

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External Links:

ISBN 0-201-13510-8, MathReview (E. Frank), ZentralBlatt

Referenced by:

§1.12(ii), §1.12(iii), §1.12(iv), §1.12(v), §13.17, §13.17, §13.5, §13.5, §4.25, §5.10.

Encodings:

BibTeX

►

Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function. In Orthogonal functions, moment theory, and continued fractions (Campinas, 1996), Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.

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External Links:

MathReview (Olav Njåstad), ZentralBlatt

Referenced by:

§5.10, §5.10.

Encodings:

BibTeX

…

36: 5.15 Polygamma Functions

… ►For $B_{2k}$ see §24.2(i). ►For continued fractions for ${\psi }^{\prime }\left(z\right)$ and ${\psi }^{\prime \prime }\left(z\right)$ see Cuyt et al. (2008, pp. 231–238).

37: 29.20 Methods of Computation

… ►A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. …

38: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series

… ►

22.12.13 $$2K\mathrm{cs}(2Kt,k)=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\frac{\pi }{\tan \left(\pi (t-n\tau )\right)}=\underset{N\to \mathrm{\infty }}{lim}\sum _{n=-N}^N{(-1)}^n\left(\underset{M\to \mathrm{\infty }}{lim}\sum _{m=-M}^M\frac{1}{t-m-n\tau }\right).$$

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Symbols:

$\mathrm{cs}(z,k)$: Jacobian elliptic function, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\tan z$: tangent function, $k$: modulus and $\tau$: ratio

Referenced by:

§22.12

Permalink:

http://dlmf.nist.gov/22.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §22.12 and Ch.22

39: 2.6 Distributional Methods

… ►

§2.6(iii) Fractional Integrals

►The Riemann–Liouville fractional integral of order $\mu$ is defined by ►

2.6.33 $${𝐼}^{\mu }f(x)=\frac{1}{\mathrm{\Gamma }\left(\mu \right)}{\int }_0^x{(x-t)}^{\mu -1}f(t)dt,$$ $\mu >0$;

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Defines:

${𝐼}^{\mu }$: fractional integral (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mu$: order and $f(t)$: locally integrable function

Referenced by:

§2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(iii), §2.6 and Ch.2

… ►

2.6.35 $${𝐼}^{\mu }f(x)=\frac{1}{\mathrm{\Gamma }\left(\mu \right)}(t^{\mu -1}\ast f)(x).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mu$: order, ${𝐼}^{\mu }$: fractional integral , $\ast$: convolution and $f(t)$: locally integrable function

Referenced by:

§2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(iii), §2.6 and Ch.2

… ►If both $f$ and $g$ in (2.6.34) have asymptotic expansions of the form (2.6.9), then the distribution method can also be used to derive an asymptotic expansion of the convolution $f\ast g$; see Li and Wong (1994). …

40: 3.1 Arithmetics and Error Measures

… ►A nonzero normalized binary floating-point machine number $x$ is represented as …where $s$ is equal to $1$ or $0$, each $b_j$, $j\ge 1$, is either $0$ or $1$, $b_1$ is the most significant bit, $p$ ($\in \mathbb{N}$) is the number of significant bits $b_j$, $b_{p-1}$ is the least significant bit, $E$ is an integer called the exponent, $b_0.b_1{b}_2\mathrm{\dots }{b}_{p-1}$ is the significand, and $f=\mathrm{.}{b}_1{b}_2\mathrm{\dots }{b}_{p-1}$ is the fractional part. …