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1: Bibliography L

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Generating Bessel functions in Mie scattering calculations using continued fractions. Applied Optics 15 (3), pp. 668–671.

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Referenced by:

§3.10(iii), §3.10(iii), Erratum (V1.0.24) for Paragraph Steed’s Algorithm (in §3.10(iii)).

Encodings:

BibTeX

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Le Calcul des Résidus et ses Applications à la Théorie des Fonctions. Gauthier-Villars, Paris (French).

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Notes:

Reprinted by Éditions Jacques Gabay, Sceaux, 1989.

External Links:

ISBN 2-87647-060-8, MathReview, ZentralBlatt

Referenced by:

(25.5.10), (25.5.11), (25.5.12).

Encodings:

BibTeX

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Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions. Numer. Math. 116 (2), pp. 269–289.

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External Links:

ISSN 0029-599X, Document, FullText, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§13.11, §13.11.

Encodings:

BibTeX

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2: 8.21 Generalized Sine and Cosine Integrals

… ►Furthermore, $\mathrm{si}(a,z)$ and $\mathrm{ci}(a,z)$ are entire functions of $a$, and $\mathrm{Si}(a,z)$ and $\mathrm{Ci}(a,z)$ are meromorphic functions of $a$ with simple poles at $a=-1,-3,-5,\mathrm{\dots }$ and $a=0,-2,-4,\mathrm{\dots }$, respectively. … ►When $\mathrm{ph}z=0$ (and when $a\ne -1,-3,-5,\mathrm{\dots }$, in the case of $\mathrm{Si}(a,z)$, or $a\ne 0,-2,-4,\mathrm{\dots }$, in the case of $\mathrm{Ci}(a,z)$) the principal values of $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). Elsewhere in the sector $|\mathrm{ph}z|\le \pi$ the principal values are defined by analytic continuation from $\mathrm{ph}z=0$; compare §4.2(i). … ►For the functions on the right-hand sides of (8.21.10) and (8.21.11) see §6.2(ii). … ►When , …

3: Customize DLMF

…You can customize the appearance and functionality of the DLMF site using these selections. …

4: 27.20 Methods of Computation: Other Number-Theoretic Functions

… ►To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). For a completely multiplicative function we use the values at the primes together with (27.3.10). ►The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function $p\left(n\right)$ for . …To compute a particular value $p\left(n\right)$ it is better to use the Hardy–Ramanujan–Rademacher series (27.14.9). … ►A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function $\tau \left(n\right)$, and the values can be checked by the congruence (27.14.20). …

5: About MathML

… ►The DLMF uses MathML Core, a subset designed specifically for browsers. ►The mathematical notations used in DLMF are complex enough that a fairly complete MathML implementation is required. …However, the MathJax javascript library can also used to render the MathML, albeit more slowly. … ►Of course you are encouraged to use a modern, up-to-date browser. … ►DLMF uses the STIX web font to provide a consistent coverage. …

6: Browsers

… ►We are trying to use the best technologies available to make the DLMF content as useful and accessible as we can. Thus, we are now using the new (almost-)standard for HTML, HTML5, as the primary format for our web content. … ►If none of those solutions work for you, you may explicitly choose a format such as HTML+images, using images for mathematics, at Customize DLMF. …

7: 27.17 Other Applications

… ►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. ►Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers. …Apostol and Zuckerman (1951) uses congruences to construct magic squares. … ►Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.

8: 21 Multidimensional Theta Functions

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9: 31 Heun Functions

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10: 5.21 Methods of Computation

… ►Or we can use forward recurrence, with an initial value obtained e. …For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). … ►Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour. …