squares and products

SearchAdvancedHelp

(0.002 seconds)

21—30 of 38 matching pages

21: Bibliography

… ►

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

ⓘ

External Links:

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

Referenced by:

§24.13(i), §24.13(i).

Encodings:

BibTeX

… ►

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

ⓘ

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

… ►

On magic squares constructed by the uniform step method. Proc. Amer. Math. Soc. 2 (4), pp. 557–565.

ⓘ

External Links:

ISSN 0002-9939, Fulltext, MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§27.17.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§10.22(iv), §14.30(ii).

Encodings:

BibTeX

…

22: Bibliography G

… ►

Recurrence relations for cross-products of Bessel functions. Quart. J. Mech. Appl. Math. 2 (1), pp. 72–74.

ⓘ

External Links:

ISSN 0033-5614, Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.6(iii).

Encodings:

BibTeX

… ►

On the exceptional zeros of cross-products of derivatives of spherical Bessel functions. Z. Angew. Math. Phys. 36 (3), pp. 491–494.

ⓘ

External Links:

ISSN 0044-2275, Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§10.58.

Encodings:

BibTeX

… ►

Table of Integrals, Series, and Products. 6th edition, Academic Press Inc., San Diego, CA.

ⓘ

Notes:

Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. For a review of the fourth edition (1980), with errata list, see Math. Comp. v. 36 (1981), pp. 310–314. For a review of the fifth edition (1994) see Math. Comp. v. 64 (1995), pp. 439–441.

External Links:

Errata in the 4th, 5th, and 6th editions, ISBN 0-12-294757-6, Link, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §1.4(iv), §1.8(v), §10.22(vi), §10.23(iv), §10.43(vi), §10.60(iv), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(vi), §13.23(v), §14.18(iv), §15.14, §18.17(ix), §18.18(ix), §19.25(v), §19.29(i), §19.29(ii), §20.10(iii), §22.14(ii), §22.14(iii), §23.14, §24.7(iii), §28.10(iii), §28.28(v), §4.10(iii), §4.11, §4.26(v), §4.27, §4.40(v), §4.41, §5.13, §6.14(iii), §7.14(iii), §8.14.

Encodings:

BibTeX

… ►

Representations of Integers as Sums of Squares. Springer-Verlag, New York.

ⓘ

External Links:

ISBN 0-387-96126-7, MathReview (Harvey Cohn), ZentralBlatt

Referenced by:

§27.13(iv).

Encodings:

BibTeX

…

23: Bibliography S

… ►

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

ⓘ

External Links:

Document, MathReview (S. Saran), ZentralBlatt

Referenced by:

§10.21(x).

Encodings:

BibTeX

… ►

On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§12.13(i).

Encodings:

BibTeX

… ►

On the expansion of the product of two parabolic cylinder functions of non integral order. Proc. Benares Math. Soc. (N. S.) 2, pp. 61–68.

ⓘ

External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

ISSN 0253-424X, MathReview

Referenced by:

§9.10(v).

Encodings:

BibTeX

►

Algorithm AS 239. Chi-squared and incomplete gamma integral. Appl. Statist. 37 (3), pp. 466–473.

ⓘ

Notes:

Double-precision Fortran.

External Links:

FullText, Code

Referenced by:

5th item.

Encodings:

BibTeX

…

24: 19.25 Relations to Other Functions

… ►

19.25.7 $$E(\varphi ,k)=2R_G(c-1,c-k^2,c)-(c-1)R_F(c-1,c-k^2,c)-\sqrt{c-1}\sqrt{c-k^2}/\sqrt{c},$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i), §19.25(i), §19.25(iii), Figure 19.3.4, Figure 19.3.4, Figure 19.3.4, §19.36(ii), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.25.E7

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.25(i), §19.25 and Ch.19

… ►

19.25.10 $$E(\varphi ,k)=k_{}^{\prime }{}_{}{}^2{R}_F(c-1,c-k^2,c)+\frac{1}{3}{k}^{2}k_{}^{\prime }{}_{}{}^2{R}_D(c-1,c,c-k^2)+k^2\sqrt{c-1}/\left(\sqrt{c}\sqrt{c-k^2}\right),$$ $c>k^2$,

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\varphi$: real or complex argument, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Referenced by:

§19.25(i), §19.30(i), §19.36(i), §19.6(iv), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.25.E10

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.25(i), §19.25 and Ch.19

►

19.25.11 $$E(\varphi ,k)=-\frac{1}{3}k_{}^{\prime }{}_{}{}^2{R}_D(c-k^2,c,c-1)+\sqrt{c-k^2}/\left(\sqrt{c}\sqrt{c-1}\right),$$ $\varphi \ne \frac{1}{2}\pi$.

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\pi$: the ratio of the circumference of a circle to its diameter, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\varphi$: real or complex argument, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Referenced by:

§19.25(i), §19.25(i), §19.36(i), §19.6(iv), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.25.E11

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.25(i), §19.25 and Ch.19

…

25: 19.20 Special Cases

… ►

19.20.19 $$R_D(x,y,z)\sim 3x^{-1/2}{y}^{-1/2}{z}^{-1/2},$$ $z/\sqrt{xy}\to 0$.

ⓘ

Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables and $\sim$: asymptotic equality

Referenced by:

§19.20(iv), Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.20.E19

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.20(iv), §19.20 and Ch.19

…

26: 1.11 Zeros of Polynomials

… ►

1.11.9

ⓘ

Defines:

$D$: discriminant of $f(z)$ (locally)

Symbols:

$z$: variable, $j$: integer, $k$: integer and $n$: nonnegative integer

Referenced by:

§1.11(ii)

Permalink:

http://dlmf.nist.gov/1.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §1.11(ii), §1.11(ii), §1.11 and Ch.1

… ►The sum and product of the roots are respectively $-b/a$ and $c/a$. … ►The square roots are chosen so that …

27: 21.7 Riemann Surfaces

… ►Either branch of the square roots may be chosen, as long as the branch is consistent across $\mathrm{\Gamma }$. … ►

21.7.17 $$\sum _{P_j\in U}\prod _{k=1}^4\theta \left[\genfrac{}{}{0.0pt}{}{{𝜶}_k+{𝜼}^1(P_j)}{{𝜷}_k+{𝜼}^2(P_j)}\right]\left({𝐳}_k\right|𝛀)=\sum _{P_j\in U^c}\prod _{k=1}^4\theta \left[\genfrac{}{}{0.0pt}{}{{𝜶}_k+{𝜼}^1(P_j)}{{𝜷}_k+{𝜼}^2(P_j)}\right]\left({𝐳}_k\right|𝛀).$$

ⓘ

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\in$: element of, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector, $𝜷$: $g$-dimensional vector, $P_j$: branch points and $U$: subset of $B$

Permalink:

http://dlmf.nist.gov/21.7.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(iii), §21.7 and Ch.21

28: 18.36 Miscellaneous Polynomials

… ►Sobolev OP’s are orthogonal with respect to an inner product involving derivatives. … ►These are matrix-valued polynomials that are orthogonal with respect to a square matrix of measures on the real line. …

29: 19.18 Derivatives and Differential Equations

… ►

19.18.6 $$\left(\frac{\partial }{\partial x}+\frac{\partial }{\partial y}+\frac{\partial }{\partial z}\right)R_F(x,y,z)=\frac{-1}{2\sqrt{x}\sqrt{y}\sqrt{z}},$$

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ and $\partial x$: partial differential of $x$

Referenced by:

§19.18(ii), §19.32, Erratum (V1.1.3) for Chapter 19

Permalink:

http://dlmf.nist.gov/19.18.E6

Encodings:

pMML, png, TeX

Correction (effective with 1.1.3):

The factors inside the square root on the right-hand side were written as products to ensure the correct multivalued behavior.

Suggested 2021-06-07 by Luc Maisonobe

See also:

Annotations for §19.18(ii), §19.18 and Ch.19

…

30: 36.12 Uniform Approximation of Integrals

… ►

36.12.8 $$a_m(𝐲)=\sum _{n=1}^{K+1}\frac{P_{mn}(𝐲)G_n(𝐲)}{{(t_n(𝐱(𝐲)))}^{m+1}\prod _{\begin{array}{c}l=1\\ l\ne n\end{array}}^{K+1}(t_n(𝐱(𝐲))-t_l(𝐱(𝐲)))},$$

ⓘ

Defines:

$a_m(𝐲)$: function (locally)

Symbols:

$l$: integer, $n$: integer, $m$: integer, $K$: codimension, $𝐱(𝐲)$: function and $t_j(𝐱)$: solutions

Permalink:

http://dlmf.nist.gov/36.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.12(i), §36.12 and Ch.36

… ►The square roots are real and positive when $𝐲$ is such that all the critical points are real, and are defined by analytic continuation elsewhere. …