integral equations and representations

SearchAdvancedHelp

(0.005 seconds)

1—10 of 128 matching pages

1: 31.10 Integral Equations and Representations

§31.10 Integral Equations and Representations

… ►

Kernel Functions

… ►For integral equations satisfied by the Heun polynomial ${\mathrm{𝐻𝑝}}_{n,m}\left(z\right)$ we have $\sigma =\frac{1}{2}-\delta -j$, $j=0,1,\mathrm{\dots },n$. … ►

Kernel Functions

… ►

2: 12.18 Methods of Computation

… ►These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …

3: 16.25 Methods of Computation

… ►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …

4: Bibliography E

… ►

A Treatise on the Integral Calculus. Vol. 1-2, Chelsea Publishing Co., New York.

ⓘ

Notes:

Originally published by Macmillan, 1921-1930.

External Links:

ZentralBlatt

Referenced by:

§19.11(i), §19.4(ii).

Encodings:

BibTeX

… ►

Integral Representation and the Computation of Combinatorial Sums. Translations of Mathematical Monographs, Vol. 59, American Mathematical Society, Providence, RI.

ⓘ

Notes:

Translated from the Russian by H. H. McFadden, Translation edited by Lev J. Leifman

External Links:

ISBN 0-8218-4512-8, MathReview, ZentralBlatt

Referenced by:

§15.17(iv).

Encodings:

BibTeX

… ►

Integral equations for Heun functions. Quart. J. Math., Oxford Ser. 13, pp. 107–112.

ⓘ

External Links:

Document, MathReview (E. Rothe), ZentralBlatt

Referenced by:

§31.10(i).

Encodings:

BibTeX

… ►

On the representations of a number as a sum of three squares. Proc. London Math. Soc. (3) 9, pp. 575–594.

ⓘ

External Links:

Document, MathReview (S. Chowla), ZentralBlatt

Referenced by:

§20.12(i).

Encodings:

BibTeX

… ►

A new series representation for $\zeta (3)$ . Amer. Math. Monthly 97 (3), pp. 219–220.

ⓘ

External Links:

ISSN 0002-9890, FullText, MathReview (Christian Radoux), ZentralBlatt

Referenced by:

(25.8.10).

Encodings:

BibTeX

…

5: Bibliography S

… ►

Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation. Adv. Quantum Chem. 72, pp. 95–127.

ⓘ

Referenced by:

§18.38(i).

Encodings:

BibTeX

… ►

On integral representations for Lamé and other special functions. SIAM J. Math. Anal. 11 (4), pp. 702–723.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (R. K. Gupta), ZentralBlatt

Referenced by:

§29.17(i), §29.8.

Encodings:

BibTeX

… ►

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.

ⓘ

External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

BibTeX

… ►

Représentation asymptotique de la solution générale de l’équation de Mathieu-Hill. Acad. Roy. Belg. Bull. Cl. Sci. (5) 51 (11), pp. 1415–1446.

ⓘ

External Links:

ZentralBlatt

Referenced by:

§28.8(ii).

Encodings:

BibTeX

… ►

Non-linear integral equations for Heun functions. Proc. Edinburgh Math. Soc. (2) 16, pp. 281–289.

ⓘ

External Links:

Document, MathReview (J. D. DePree), ZentralBlatt

Referenced by:

§31.10(ii).

Encodings:

BibTeX

…

6: 6.7 Integral Representations

§6.7 Integral Representations

… ► ►

§6.7(ii) Sine and Cosine Integrals

… ►

§6.7(iii) Auxiliary Functions

… ►For collections of integral representations see Bierens de Haan (1939, pp. 56–59, 72–73, 82–84, 121, 133–136, 155, 179–181, 223, 225–227, 230, 259–260, 374, 377, 397–398, 408, 416, 424, 431, 438–439, 442–444, 488, 496–500, 567–571, 585, 602, 638, 675–677), Corrington (1961), Erdélyi et al. (1954a, vol. 1, pp. 267–270), Geller and Ng (1969), Nielsen (1906b), Oberhettinger (1974, pp. 244–246), Oberhettinger and Badii (1973, pp. 364–371), and Watrasiewicz (1967).

7: 28.28 Integrals, Integral Representations, and Integral Equations

§28.28 Integrals, Integral Representations, and Integral Equations

►

§28.28(i) Equations with Elementary Kernels

… ►

28.28.23 $$\frac{2}{\pi }{\int }_0^{\pi }{𝒞}_{2\mathrm{\ell }+2}^{(j)}(2hR)\sin \left((2\mathrm{\ell }+2)\varphi \right){\mathrm{se}}_{2m+2}(t,h^2)dt={(-1)}^{\mathrm{\ell }+m}{B}_{2\mathrm{\ell }+2}^{2m+2}(h^2){\mathrm{Ms}}_{2m+2}^{(j)}(z,h).$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $\sin z$: sine function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions, $\varphi (z,t)$: function, $R(z,t)$: function and $B_m(q)$: Fourier coefficient

A&S Ref:

20.7.28 (in different form and only for $\mathrm{\ell }=0$)

Permalink:

http://dlmf.nist.gov/28.28.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(ii), §28.28 and Ch.28

… ►

28.28.49 $${\widehat{\alpha }}_{n,m}^{(c)}=\frac{1}{2\pi }{\int }_0^{2\pi }\cos t{\mathrm{ce}}_n(t,h^2){\mathrm{ce}}_m(t,h^2)dt={(-1)}^{p+1}\frac{2}{\mathrm{i}\pi }\frac{{\mathrm{ce}}_n(0,h^2){\mathrm{ce}}_m(0,h^2)}{h{\mathrm{Dc}}_0(n,m,0)}.$$

ⓘ

Defines:

${\widehat{\alpha }}_{n,m}$ (locally)

Symbols:

${\mathrm{Dc}}_j(n,m,z)$: cross-products of radial Mathieu functions and their derivatives, ${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $m$: integer, $h$: parameter and $n$: integer

Permalink:

http://dlmf.nist.gov/28.28.E49

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(iv), §28.28 and Ch.28

…

8: 9.11 Products

… ►

§9.11(iii) Integral Representations

… ►For an integral representation of the Dirac delta involving a product of two $\mathrm{Ai}$ functions see §1.17(ii). ►For further integral representations see Reid (1995, 1997a, 1997b). ►

§9.11(iv) Indefinite Integrals

… ►

§9.11(v) Definite Integrals

…

9: Bibliography K

… ►

On an integral of Hardy and Littlewood. Kyushu J. Math. 52 (1), pp. 249–263.

ⓘ

External Links:

ISSN 1340-6116, Document, MathReview (Fritz Haslinger), ZentralBlatt

Referenced by:

§9.13(ii).

Encodings:

BibTeX

… ►

Integral equations for special functions of Heun class. Methods Appl. Anal. 3 (4), pp. 447–456.

ⓘ

External Links:

ISSN 1073-2772, FullText, MathReview, ZentralBlatt

Referenced by:

§31.10(ii).

Encodings:

BibTeX

… ►

Closed-form representations of the Lambert $W$ function. Fract. Calc. Appl. Anal. 7 (2), pp. 177–190.

ⓘ

External Links:

ISSN 1311-0454, MathReview (Lewis H. Ryder), ZentralBlatt

Referenced by:

§4.13.

Encodings:

BibTeX

… ►

Rational solutions of the fifth Painlevé equation. Differential Integral Equations 7 (3-4), pp. 967–1000.

ⓘ

External Links:

ISSN 0893-4983, MathReview, ZentralBlatt

Referenced by:

§32.8(v).

Encodings:

BibTeX

… ►

Compact quantum groups and $q$-special functions. In Representations of Lie Groups and Quantum Groups, Pitman Res. Notes Math. Ser., Vol. 311, pp. 46–128.

ⓘ

Notes:

Sections 3 and 4 also available as arXiv:math/9403216

External Links:

MathReview (Eric M. Opdam)

Referenced by:

(18.27.14_3).

Encodings:

BibTeX

…

10: 7.7 Integral Representations

… ►

7.7.9 $${\int }_0^x\mathrm{erf}tdt=x\mathrm{erf}x+\frac{1}{\sqrt{\pi }}\left({\mathrm{e}}^{-x^2}-1\right).$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $x$: real variable

A&S Ref:

7.4.35 (in different form)

Referenced by:

§7.7(i)

Permalink:

http://dlmf.nist.gov/7.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §7.7(i), §7.7 and Ch.7

…