De Moivre theorem

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21: 2 Asymptotic Approximations

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22: 6.17 Physical Applications

… ►For applications in astrophysics, see also van de Hulst (1980). …

23: 34 3j, 6j, 9j Symbols

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24: Javier Segura

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25: Bibliography B

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Periods for Calabi-Yau and Landau-Ginzburg vacua. Nuclear Phys. B 419 (2), pp. 352–403.

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External Links:

ISSN 0550-3213, Document, MathReview (Bruce Hunt), ZentralBlatt

Referenced by:

§16.23(i).

Encodings:

BibTeX

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Nouvelles Tables d’Intégrales Définies. P. Engels, Leide.

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Notes:

Corrected edition printed by Hafner Press, 1965.

Referenced by:

§1.4(iv).

Encodings:

BibTeX

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Nouvelles Tables d’Intégrales Définies. G.E. Stechert & Co., New York.

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Referenced by:

§4.10(iii), §4.26(v), §4.40(v), §6.14(iii), §6.7(iv).

Encodings:

BibTeX

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Recherches sur les transcendantes de M. Painlevé et l’étude asymptotique des équations différentielles du second ordre. Ann. Sci. École Norm. Sup. (3) 30, pp. 255–375.

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External Links:

ISSN 0012-9593, MathReview Entry, ZentralBlatt

Referenced by:

§32.12(i), §32.12(ii).

Encodings:

BibTeX

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Interpolation und genäherte Quadratur. In Mathematische Hilfsmittel des Ingenieurs. Teil III, R. Sauer and I. Szabó (Eds.), Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, Vol. 141, pp. 232–319.

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External Links:

MathReview (E. Kreyszig)

Referenced by:

§3.3(vi).

Encodings:

BibTeX

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26: 2.7 Differential Equations

… ►An ordinary point of the differential equation … ►For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows: ►

Liouville–Green Approximation Theorem

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2.7.36 $$w(z)=A{w}_1(z)+B{w}_2(z),$$

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Symbols:

$w$: DE solution, $A$: constant, $B$: constant and $w_j(z)$: solutions

Permalink:

http://dlmf.nist.gov/2.7.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(iv), §2.7 and Ch.2

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2.7.37 $$w(z)=C{w}_3(z)+D{w}_4(z),$$

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Symbols:

$w$: DE solution, $C$: constant, $D$: constant and $w_j(z)$: solutions

Permalink:

http://dlmf.nist.gov/2.7.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §2.7(iv), §2.7 and Ch.2

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27: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod $m_1$), (mod $m_2$), (mod $m_3$), and (mod $m_4$), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result $(modm)$, which is correct to 20 digits. …

28: 1.4 Calculus of One Variable

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Mean Value Theorem

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Fundamental Theorem of Calculus

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First Mean Value Theorem

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Second Mean Value Theorem

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§1.4(vi) Taylor’s Theorem for Real Variables

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29: 34.5 Basic Properties: $\mathit{6}j$ Symbol

… ►They constitute addition theorems for the $\mathit{6}j$ symbol. …

30: 9.16 Physical Applications

… ►Details of the Airy theory are given in van de Hulst (1957) in the chapter on the optics of a raindrop. … ►An application of Airy functions to the solution of this equation is given in Gramtcheff (1981). … ►These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation). …