fundamental theorem of arithmetic

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1: 19.8 Quadratic Transformations

… ►

§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)

… ►As

n\to\infty

a_{n}

and

g_{n}

converge to a common limit

M\left(a_{0},g_{0}\right)

called the AGM (Arithmetic-Geometric Mean) of

a_{0}

and

g_{0}

. …showing that the convergence of

c_{n}

to 0 and of

a_{n}

and

g_{n}

M\left(a_{0},g_{0}\right)

is quadratic in each case. … ►

19.8.5

K\left(k\right)=\frac{\pi}{2M\left(1,k^{\prime}\right)},

-\infty<k^{2}<1

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Symbols:: $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.8.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.8(i), §19.8 and Ch.19

… ►Again,

p_{n}

and

\varepsilon_{n}

converge quadratically to

M\left(a_{0},g_{0}\right)

and 0, respectively, and

Q_{n}

converges to 0 faster than quadratically. …

2: 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, … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) ►This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the prime number theorem. … ►If

\left(a,n\right)=1

, then the Euler–Fermat theorem states that …

3: 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),

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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).

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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

…

4: 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. …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. Because each residue has no more than five digits, the arithmetic can be performed efficiently on these residues with respect to each of the moduli, yielding answers

a_{1}\pmod{m_{1}}

a_{2}\pmod{m_{2}}

a_{3}\pmod{m_{3}}

, and

a_{4}\pmod{m_{4}}

, where each

a_{j}

has no more than five digits. These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result

\pmod{m}

, which is correct to 20 digits. …

5: 27.4 Euler Products and Dirichlet Series

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

6: 28.2 Definitions and Basic Properties

… ►(28.2.1) possesses a fundamental pair of solutions

w_{\mbox{\tiny I}}(z;a,q),w_{\mbox{\tiny II}}(z;a,q)

called basic solutions with … ►

28.2.6

\mathscr{W}\left\{w_{\mbox{\tiny I}},w_{\mbox{\tiny II}}\right\}=1,

ⓘ

Symbols:: $\mathscr{W}$ : Wronskian, $w_{\mbox{\tiny I}}(z;a,q)$ : fundamental solution and $w_{\mbox{\tiny II}}(z;a,q)$ : fundamental solution
Referenced by:: §28.29(ii)
Permalink:: http://dlmf.nist.gov/28.2.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §28.2(ii), §28.2 and Ch.28

… ►

§28.2(iii) Floquet’s Theorem and the Characteristic Exponents

… ►If

q\neq 0

, then for a given value of

\nu

the corresponding Floquet solution is unique, except for an arbitrary constant factor (Theorem of Ince; see also 28.5(i)). …

7: Bibliography C

… ►

L. Carlitz (1961b) The Staudt-Clausen theorem. Math. Mag. 34, pp. 131–146.

ⓘ

External Links:: MathReview (K. Goldberg), ZentralBlatt
Referenced by:: §24.6.
Encodings:: BibTeX

… ►

C. W. Clenshaw, F. W. J. Olver, and P. R. Turner (1989) Level-Index Arithmetic: An Introductory Survey. In Numerical Analysis and Parallel Processing (Lancaster, 1987), P. R. Turner (Ed.), Lecture Notes in Math., Vol. 1397, pp. 95–168.

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External Links:: MathReview Entry, ZentralBlatt
Referenced by:: §3.1(iv).
Encodings:: BibTeX

… ►

H. S. Cohl (2013a) Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems. SIGMA Symmetry Integrability Geom. Methods Appl. 9, pp. Paper 042, 26.

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External Links:: ISSN 1815-0659, Document, MathReview (Heinrich Begehr), ZentralBlatt
Referenced by:: §14.28(ii), §14.28(ii).
Encodings:: BibTeX

… ►

D. A. Cox (1984) The arithmetic-geometric mean of Gauss. Enseign. Math. (2) 30 (3-4), pp. 275–330.

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External Links:: ISSN 0013-8584, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §19.8(i).
Encodings:: BibTeX

►

D. A. Cox (1985) Gauss and the arithmetic-geometric mean. Notices Amer. Math. Soc. 32 (2), pp. 147–151.

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External Links:: ISSN 0002-9920, MathReview (Willard Parker)
Referenced by:: §19.8(i).
Encodings:: BibTeX

…

8: 28.27 Addition Theorems

§28.27 Addition Theorems

►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …

9: Bibliography S

… ►

K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:: ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §22.19(i), §22.20(vi).
Encodings:: BibTeX

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J.-P. Serre (1973) A Course in Arithmetic. Graduate Texts in Mathematics, Vol. 7, Springer-Verlag, New York.

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Notes:: Translated from the French.
External Links:: MathReview, ZentralBlatt
Referenced by:: §20.12(i), §20.7(viii), §20.9(i), §20.9(ii), §22.18(iv), §23.15(i), §23.18, §23.19, §23.20(v).
Encodings:: BibTeX

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I. Sh. Slavutskiĭ (1995) Staudt and arithmetical properties of Bernoulli numbers. Historia Sci. (2) 5 (1), pp. 69–74.

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External Links:: ISSN 0285-4821, MathReview (Peter M. Neumann), ZentralBlatt
Referenced by:: §24.10(ii).
Encodings:: BibTeX

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D. M. Smith (1998) Algorithm 786: Multiple-precision complex arithmetic and functions. ACM Trans. Math. Software 24 (4), pp. 359–367.

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External Links:: ISSN 0098-3500, Document, GAMS (module 786; package TOMS), ZentralBlatt
Referenced by:: 4th item.
Encodings:: BibTeX

… ►

F. C. Smith (1939b) Relations among the fundamental solutions of the generalized hypergeometric equation when

p=q+1

. II. Logarithmic cases. Bull. Amer. Math. Soc. 45 (12), pp. 927–935.

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External Links:: Document, MathReview (C. S. Meijer), ZentralBlatt
Referenced by:: §16.8(ii).
Encodings:: BibTeX

…

10: Bibliography K

… ►

R. B. Kearfott (1996) Algorithm 763: INTERVAL_ARITHMETIC: A Fortran 90 module for an interval data type. ACM Trans. Math. Software 22 (4), pp. 385–392.

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External Links:: ISSN 0098-3500, Document, GAMS (module 763; package TOMS), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

… ►

Y. S. Kim, A. K. Rathie, and R. B. Paris (2013) An extension of Saalschütz’s summation theorem for the series

{}_{r+3}F_{r+2}

. Integral Transforms Spec. Funct. 24 (11), pp. 916–921.

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External Links:: ISSN 1065-2469, Document, FullText, MathReview (Christian Lavault), ZentralBlatt
Referenced by:: §16.4(ii), §16.4(ii).
Encodings:: BibTeX

… ►

B. J. King and A. L. Van Buren (1973) A general addition theorem for spheroidal wave functions. SIAM J. Math. Anal. 4 (1), pp. 149–160.

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External Links:: Document, MathReview (J. Meixner), ZentralBlatt
Referenced by:: §30.10.
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

… ►

T. H. Koornwinder (1975a) A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.

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External Links:: Document, MathReview (G. Gasper), ZentralBlatt
Referenced by:: §18.10(i).
Encodings:: BibTeX

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