fundamental theorem of calculus

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11: 24.17 Mathematical Applications

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Calculus of Finite Differences

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§24.17(iii) Number Theory

►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and $L$-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); $p$-adic analysis (Koblitz (1984, Chapter 2)). …

12: 27.4 Euler Products and Dirichlet Series

… ►The fundamental theorem of arithmetic is linked to analysis through the concept of the Euler product. …

13: Leonard C. Maximon

… ►Maximon published numerous papers on the fundamental processes of quantum electrodynamics and on the special functions of mathematical physics. …

14: Bibliography R

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Normal limit theorems for symmetric random matrices. Probab. Theory Related Fields 112 (3), pp. 411–423.

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External Links:

ISSN 0178-8051, Document, MathReview (Alexei Khorunzhy), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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13 Lectures on Fermat’s Last Theorem. Springer-Verlag, New York.

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External Links:

ISBN 0-387-90432-8, MathReview (T. Metsänkylä), ZentralBlatt

Referenced by:

§24.10(iv), §24.17(iii).

Encodings:

BibTeX

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The umbral calculus. Pure and Applied Mathematics, Vol. 111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York.

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External Links:

ISBN 0-12-594380-6, MathReview (R. Askey)

Referenced by:

§18.2(xii).

Encodings:

BibTeX

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On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42, pp. 684–760.

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External Links:

ISSN 0022-247X, Document, Link, MathReview (P. Saphar), ZentralBlatt

Referenced by:

§18.2(xii).

Encodings:

BibTeX

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Dawson’s integral and the sampling theorem. Computers in Physics 3 (2), pp. 85–87.

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Referenced by:

§7.25(vi).

Encodings:

BibTeX

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15: 1.5 Calculus of Two or More Variables

§1.5 Calculus of Two or More Variables

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Implicit Function Theorem

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§1.5(iii) Taylor’s Theorem; Maxima and Minima

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§1.5(iv) Leibniz’s Theorem for Differentiation of Integrals

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16: 17.2 Calculus

§17.2 Calculus

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§17.2(i) $q$-Calculus

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§17.2(iii) Binomial Theorem

… ►In the limit as $q\to 1$, (17.2.35) reduces to the standard binomial theorem … ►

17: 1.9 Calculus of a Complex Variable

§1.9 Calculus of a Complex Variable

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DeMoivre’s Theorem

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Cauchy’s Theorem

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Liouville’s Theorem

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Dominated Convergence Theorem

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18: 27.14 Unrestricted Partitions

… ►A fundamental problem studies the number of ways $n$ can be written as a sum of positive integers $\le n$, that is, the number of solutions of … ►Euler’s pentagonal number theorem states that …

19: 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. …

20: 24.4 Basic Properties

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Raabe’s Theorem

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24.4.39 $$E_n\left(x+h\right)={(E(x)+h)}^n.$$

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $n$: integer and $x$: real or complex

A&S Ref:

23.1.26

Permalink:

http://dlmf.nist.gov/24.4.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(viii), §24.4 and Ch.24

►For these results and also connections with the umbral calculus see Gessel (2003). …