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1: 27.2 Functions

… ►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, ►

27.2.1 $$n=\prod _{r=1}^{\nu \left(n\right)}{p}_r^{a_r},$$

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Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a$: primitive root

Referenced by:

§27.2(i), §27.3

Permalink:

http://dlmf.nist.gov/27.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►It is the special case $k=2$ of the function $d_k\left(n\right)$ that counts the number of ways of expressing $n$ as the product of $k$ factors, with the order of factors taken into account. … ►

Table 27.2.2: Functions related to division.

► ►►►

$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
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7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
…

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2: Bibliography R

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Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions. ACM Trans. Math. Softw. 40 (2), pp. 14:1–14:12.

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External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii), 7th item, §10.77(ix).

Encodings:

BibTeX

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Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

(9.11.16), (9.11.17), §9.11(iii), §9.11(v).

Encodings:

BibTeX

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Integral representations for products of Airy functions. III. Quartic products. Z. Angew. Math. Phys. 48 (4), pp. 656–664.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v).

Encodings:

BibTeX

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Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).

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External Links:

Link, ISSN 2227-7390, Document

Referenced by:

§8.15.

Encodings:

BibTeX

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Sources in the development of mathematics. Cambridge University Press, Cambridge.

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Notes:

Infinite Series and Products from the Fifteenth to the Twenty-first Century

External Links:

ISBN 978-0-521-11470-7, Document, Link, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

Profile Ranjan Roy.

Encodings:

BibTeX

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3: Bibliography D

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Sums of products of Bernoulli numbers. J. Number Theory 60 (1), pp. 23–41.

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External Links:

ISSN 0022-314X, Document, MathReview (Wen Peng Zhang), ZentralBlatt

Referenced by:

§24.14(ii).

Encodings:

BibTeX

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Infinite integrals in the theory of Bessel functions. Quart. J. Math., Oxford Ser. 1 (1), pp. 122–145.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.32(iii).

Encodings:

BibTeX

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Vector coupling coefficients as products of prime factors. Comput. Phys. Comm. 4 (2), pp. 268–274.

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Notes:

Single-precision Fortran

External Links:

Document, Code

Referenced by:

4th item.

Encodings:

BibTeX

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal. 9 (1), pp. 76–86.

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External Links:

Document, MathReview (R. C. Varma), ZentralBlatt

Referenced by:

§18.17(iii).

Encodings:

BibTeX

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4: Bibliography S

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A property of the zeros of cross-product Bessel functions of different orders. Z. Angew. Math. Mech. 56 (2), pp. 120–121.

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External Links:

Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§10.21(x).

Encodings:

BibTeX

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On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

BibTeX

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:

Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.

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External Links:

ISSN 0253-424X, MathReview

Referenced by:

§9.10(v).

Encodings:

BibTeX

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Computation of infinite integrals involving Bessel functions of arbitrary order by the $\overline{D}$-transformation. J. Comput. Appl. Math. 78 (1), pp. 125–130.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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5: Bibliography C

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On the representation of a large even integer as the sum of a prime and the product of at most two primes. Kexue Tongbao (Foreign Lang. Ed.) 17, pp. 385–386.

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External Links:

MathReview (T. Hayashida)

Referenced by:

§27.13(ii).

Encodings:

BibTeX

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Remarks on the zeros of cross-product Bessel functions. J. Soc. Indust. Appl. Math. 12 (3), pp. 580–587.

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External Links:

Document, MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§10.21(x), §10.21(x).

Encodings:

BibTeX

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The analyticity of cross-product Bessel function zeros. Proc. Cambridge Philos. Soc. 62, pp. 215–226.

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External Links:

MathReview (R. G. Langebartel), ZentralBlatt

Referenced by:

§10.21(x).

Encodings:

BibTeX

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The asymptotic nature of zeros of cross-product Bessel functions. Quart. J. Mech. Appl. Math. 19 (4), pp. 511–522.

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External Links:

ISSN 0033-5614, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.21(x).

Encodings:

BibTeX

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Product formulas and convolutions for angular and radial spheroidal wave functions. Trans. Amer. Math. Soc. 338 (2), pp. 695–710.

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External Links:

ISSN 0002-9947, FullText, MathReview (Jean-Claude Lucquiaud), ZentralBlatt

Referenced by:

§30.10, §30.11(vi).

Encodings:

BibTeX

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6: Bibliography I

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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of $J_0(z)-i{J}_1(z)$ and of Bessel functions $J_m(z)$ of any real order $m$ . Linear Algebra Appl. 194, pp. 35–70.

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External Links:

ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

BibTeX

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Computation of integrals over the half-line involving products of Bessel functions, with application to microwave transmission lines. Z. Angew. Math. Mech. 75 (12), pp. 917–926.

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External Links:

ISSN 0044-2267, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

BibTeX

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

Encodings:

BibTeX

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On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.

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External Links:

ISSN 0021-9045, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.36(ii).

Encodings:

BibTeX

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7: Bibliography

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Integrals of products of Bernoulli polynomials. J. Math. Anal. Appl. 381 (1), pp. 10–16.

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External Links:

ISSN 0022-247X, Document, FullText, MathReview (Pablo Manuel Román), ZentralBlatt

Referenced by:

§24.13(i), §24.13(i).

Encodings:

BibTeX

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Integrals of products of Airy functions. J. Phys. A 10 (4), pp. 485–490.

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External Links:

Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

(9.11.10), (9.11.5), (9.11.6), (9.11.7), (9.11.8), (9.11.9), §9.11(iv), §9.11(iv).

Encodings:

BibTeX

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There are infinitely many Carmichael numbers. Ann. of Math. (2) 139 (3), pp. 703–722.

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External Links:

ISSN 0003-486X, FullText, MathReview (H. L. Montgomery), ZentralBlatt

Referenced by:

§27.12.

Encodings:

BibTeX

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Table of Definite and Infinite Integrals. Physical Sciences Data, Vol. 13, Elsevier Scientific Publishing Co., Amsterdam.

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Notes:

Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1386.

External Links:

ISBN 0-444-42151-3, MathReview (R. P. Boas), ZentralBlatt

Referenced by:

§1.4(iv), §10.22(vi), §10.43(vi), §11.7(v), §11.9(iv), §12.5(iv), §13.10(vi), §13.23(v), §15.14, §16.20, §16.5, §18.17(ix), §4.10(iii), §4.26(v), §4.40(v), §5.13, §6.14(iii), §7.14(iii), §8.14, §8.19(x).

Encodings:

BibTeX

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An integral of products of ultraspherical functions and a $q$-extension. J. London Math. Soc. (2) 33 (1), pp. 133–148.

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External Links:

ISSN 0024-6107, MathReview (David M. Bressoud), ZentralBlatt

Referenced by:

§10.22(iv), §14.30(ii).

Encodings:

BibTeX

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8: Bibliography W

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Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57 (196), pp. 723–733.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§4.12.

Encodings:

BibTeX

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Reduction formulae for products of theta functions. J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.

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External Links:

Document

Referenced by:

(20.7.34), §20.7(ix).

Encodings:

BibTeX

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:

ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

Encodings:

BibTeX

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The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A, pp. 32–52.

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External Links:

FullText, Document, ZentralBlatt

Referenced by:

§35.3(i), §35.3(ii).

Encodings:

BibTeX

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9: Bibliography V

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Integrating products of Bessel functions with an additional exponential or rational factor. Comput. Phys. Comm. 178 (8), pp. 578–590.

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Notes:

Associated computer program available online

External Links:

ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

Encodings:

BibTeX

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An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.

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External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

BibTeX

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Some novel infinite series of spherical Bessel functions. Quart. Appl. Math. 42 (3), pp. 321–324.

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External Links:

ISSN 0033-569X, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.60(iii), §10.60(iii).

Encodings:

BibTeX

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Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

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External Links:

ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt

Referenced by:

§31.15(iii).

Encodings:

BibTeX

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Integral representations for products of Lamé functions by use of fundamental solutions. SIAM J. Math. Anal. 15 (3), pp. 559–569.

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External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8.

Encodings:

BibTeX

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