fundamental%20theorem%20of%20calculus

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1: 27.2 Functions

… ►

§27.2(i) Definitions

►Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer

n>1

can be represented uniquely as a product of prime powers, …Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ►If

\left(a,n\right)=1

, then the Euler–Fermat theorem states that …

2: 27.15 Chinese Remainder Theorem

§27.15 Chinese Remainder Theorem

… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product

m

has 20 digits, twice the number of digits in the data. 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

\pmod{m}

, which is correct to 20 digits. …

3: 20 Theta Functions

Chapter 20 Theta Functions

…

4: Bibliography K

… ►

R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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External Links:: ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:: Document, MathReview (Matti Jutila), ZentralBlatt
Referenced by:: §25.18(i).
Encodings:: BibTeX

… ►

A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:: ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

… ►

D. E. Knuth (1968) The Art of Computer Programming. Vol. 1: Fundamental Algorithms. 1st edition, Addison-Wesley Publishing Co., Reading, MA-London-Don Mills, Ont.

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Notes:: Second printing of second edition by Addison-Wesley Publishing Co., Reading, MA-London-Amsterdam, 1975.
External Links:: MathReview (M. Muller), ZentralBlatt
Referenced by:: §1.4(iii), (25.11.33).
Encodings:: BibTeX

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T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:: ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt
Referenced by:: §18.26(v), §18.26(v).
Encodings:: BibTeX

…

5: Bibliography M

… ►

A. J. MacLeod (1996b) Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:: Includes Fortran code, maximum accuracy 20S.
External Links:: ISSN 1017-1398, Document, MathReview, ZentralBlatt
Referenced by:: §6.13, 4th item, §6.21(ii).
Encodings:: BibTeX

… ►

Fr. Mechel (1966) Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:: ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt
Referenced by:: §10.74(iii).
Encodings:: BibTeX

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R. Metzler, J. Klafter, and J. Jortner (1999) Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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External Links:: FullText
Referenced by:: §8.24(i).
Encodings:: BibTeX

… ►

D. S. Moak (1981) The

q

-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:: ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.27(v).
Encodings:: BibTeX

… ►

P. J. Mohr and B. N. Taylor (2005) CODATA recommended values of the fundamental physical constants: 2002. Rev. Mod.Phys. 77, pp. 1–107.

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Referenced by:: §18.39(ii).
Encodings:: BibTeX

…

6: Bibliography G

… ►

W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:: For remark see same journal 24(1998), p. 355.
External Links:: Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt
Referenced by:: 2nd item, §3.5(v), §3.5(v).
Encodings:: BibTeX

… ►

A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:: Includes Fortran program claiming 12S accuracy
External Links:: ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt
Referenced by:: 3rd item, §10.77(ix).
Encodings:: BibTeX

… ►

K. Girstmair (1990a) A theorem on the numerators of the Bernoulli numbers. Amer. Math. Monthly 97 (2), pp. 136–138.

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External Links:: ISSN 0002-9890, FullText, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.10(iv), §24.16(iii).
Encodings:: BibTeX

… ►

K. Gottfried and T. Yan (2004) Quantum mechanics: fundamentals. Second edition, Springer-Verlag, New York.

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External Links:: ISBN 0-387-22023-2, MathReview (Howard E. Brandt), ZentralBlatt
Referenced by:: §18.39(i).
Encodings:: BibTeX

… ►

Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:: ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

…

7: Tom M. Apostol

… ►Apostol was born on August 20, 1923. … ►He was internationally known for his textbooks on calculus, analysis, and analytic number theory, which have been translated into five languages, and for creating Project MATHEMATICS!, a series of video programs that bring mathematics to life with computer animation, live action, music, and special effects. … ►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). …

8: Peter L. Walker

… ►Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. … ►

20 Theta Functions

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9: 28.29 Definitions and Basic Properties

… ►

§28.29(ii) Floquet’s Theorem and the Characteristic Exponent

… ►

28.29.4

w_{\mbox{\tiny I}}(z+\pi,\lambda)=w_{\mbox{\tiny I}}(\pi,\lambda)w_{\mbox{% \tiny I}}(z,\lambda)+w^{\prime}_{\mbox{\tiny I}}(\pi,\lambda)w_{\mbox{\tiny II% }}(z,\lambda),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex variable, $w(z)$ : Mathieu’s equation solution, $w_{\mbox{\tiny I}}(z;a,q)$ : fundamental solution, $w_{\mbox{\tiny II}}(z;a,q)$ : fundamental solution and $\lambda$ : parameter
Permalink:: http://dlmf.nist.gov/28.29.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §28.29(ii), §28.29 and Ch.28

►

28.29.5

w_{\mbox{\tiny II}}(z+\pi,\lambda)=w_{\mbox{\tiny II}}(\pi,\lambda)w_{\mbox{% \tiny I}}(z,\lambda)+w^{\prime}_{\mbox{\tiny II}}(\pi,\lambda)w_{\mbox{\tiny II% }}(z,\lambda).

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $z$ : complex variable, $w(z)$ : Mathieu’s equation solution, $w_{\mbox{\tiny I}}(z;a,q)$ : fundamental solution, $w_{\mbox{\tiny II}}(z;a,q)$ : fundamental solution and $\lambda$ : parameter
Permalink:: http://dlmf.nist.gov/28.29.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §28.29(ii), §28.29 and Ch.28

… ►If

\nu

(\neq 0,1)

is a solution of (28.29.9), then

F_{\nu}(z)

F_{-\nu}(z)

comprise a fundamental pair of solutions of Hill’s equation. … ►

28.29.15

\bigtriangleup(\lambda)=w_{\mbox{\tiny I}}(\pi,\lambda)+w_{\mbox{\tiny II}}^{% \prime}(\pi,\lambda)

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $w_{\mbox{\tiny I}}(z;a,q)$ : fundamental solution, $w_{\mbox{\tiny II}}(z;a,q)$ : fundamental solution and $\lambda$ : parameter
Permalink:: http://dlmf.nist.gov/28.29.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §28.29(iii), §28.29 and Ch.28

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10: 8 Incomplete Gamma and Related
Functions

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