Weber–Schafheitlin discontinuous integrals

SearchAdvancedHelp

(0.002 seconds)

31—40 of 455 matching pages

31: Bibliography S

… ►

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

►

On certain integrals and expansions involving Weber’s parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 4, pp. 158–166.

ⓘ

External Links:

MathReview (M. C. Gray), ZentralBlatt

Referenced by:

§12.13(ii).

Encodings:

BibTeX

… ►

Gaussian-transform method for molecular integrals. I. Formulation for energy integrals. J. Chem. Phys. 43 (2), pp. 398–414.

ⓘ

External Links:

Document, MathReview (R. K. Nesbet)

Referenced by:

§8.24(i).

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

… ►

Quelques intégrales définies discontinues contenant des fonctions de Mathieu. Acad. Roy. Belg. Bull. Cl. Sci. (5) 56 (5), pp. 475–491 (French).

ⓘ

External Links:

MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§28.28(i), §28.28(v).

Encodings:

BibTeX

…

32: 2.8 Differential Equations with a Parameter

… ►For $J_{\nu }$ and $Y_{\nu }$ see §10.2(ii). … ►

2.8.32 $$J_{\nu }(x)+Y_{\nu }(x)=0.$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind and $\nu$: real nonnegative constant

Permalink:

http://dlmf.nist.gov/2.8.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iv), §2.8 and Ch.2

►Define … ►

2.8.34 $$\mathrm{env}{J}_{\nu }(x)=\mathrm{env}{Y}_{\nu }(x)={\left(J_{\nu }^2(x)+Y_{\nu }^2(x)\right)}^{1/2},$$ .

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{env}{J}_{\nu }\left(x\right)$: envelope of Bessel function $J_{\nu }\left(x\right)$ , $\mathrm{env}{Y}_{\nu }\left(x\right)$: envelope of Bessel function $Y_{\nu }\left(x\right)$ , $\nu$: real nonnegative constant and $X_{\nu }$: smallest positive root

Permalink:

http://dlmf.nist.gov/2.8.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §2.8(iv), §2.8 and Ch.2

… ►Corresponding to the problems for integrals outlined in §§2.3(v), 2.4(v), and 2.4(vi), there are analogous problems for differential equations. …

33: Bibliography N

… ►

Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

ⓘ

External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

Encodings:

BibTeX

… ►

The resurgence properties of the large order asymptotics of the Anger-Weber function I. J. Class. Anal. 4 (1), pp. 1–39.

ⓘ

External Links:

ISSN 1848-5979, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

BibTeX

►

The resurgence properties of the large order asymptotics of the Anger-Weber function II. J. Class. Anal. 4 (2), pp. 121–147.

ⓘ

External Links:

ISSN 1848-5979, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

BibTeX

… ►

Elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 99–102.

ⓘ

Notes:

Includes Algol program.

External Links:

MathReview

Referenced by:

§19.39(iii).

Encodings:

BibTeX

►

On the calculation of elliptic integrals of the second and third kinds. Zastos. Mat. 11, pp. 91–94.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(ii).

Encodings:

BibTeX

…

34: 11.9 Lommel Functions

… ►

11.9.2 $$w=s_{\mu ,\nu }\left(z\right)+A{J}_{\nu }\left(z\right)+B{Y}_{\nu }\left(z\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $s_{\mu ,\nu }\left(z\right)$: Lommel function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Permalink:

http://dlmf.nist.gov/11.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(i), §11.9 and Ch.11

… ►

11.9.5 $$S_{\mu ,\nu }\left(z\right)=s_{\mu ,\nu }\left(z\right)+2^{\mu -1}\mathrm{\Gamma }\left(\frac{1}{2}\mu +\frac{1}{2}\nu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\frac{1}{2}\mu -\frac{1}{2}\nu +\frac{1}{2}\right)\left(\sin \left(\frac{1}{2}(\mu -\nu )\pi \right)J_{\nu }\left(z\right)-\cos \left(\frac{1}{2}(\mu -\nu )\pi \right)Y_{\nu }\left(z\right)\right),$$

ⓘ

Defines:

$S_{\mu ,\nu }\left(z\right)$: Lommel function

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $s_{\mu ,\nu }\left(z\right)$: Lommel function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function, $z$: complex variable and $\nu$: real or complex order

Permalink:

http://dlmf.nist.gov/11.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(i), §11.9 and Ch.11

… ►For collections of integral representations and integrals see Apelblat (1983, §12.17), Babister (1967, p. 85), Erdélyi et al. (1954a, §§4.19 and 5.17), Gradshteyn and Ryzhik (2000, §6.86), Marichev (1983, p. 193), Oberhettinger (1972, pp. 127–128, 168–169, and 188–189), Oberhettinger (1974, §§1.12 and 2.7), Oberhettinger (1990, pp. 105–106 and 191–192), Oberhettinger and Badii (1973, §2.14), Prudnikov et al. (1990, §§1.6 and 2.9), Prudnikov et al. (1992a, §3.34), and Prudnikov et al. (1992b, §3.32).

35: Bibliography B

… ►

A note on integrals involving parabolic cylinder functions. SIAM J. Appl. Math. 16 (1), pp. 71–74.

ⓘ

External Links:

Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§12.12.

Encodings:

BibTeX

… ►

Numerical calculation of a generalized complete elliptic integral. Rev. Mod. Phys. 10, pp. 264–269.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§19.2(iii).

Encodings:

BibTeX

… ►

Tables of the Anger and Lommel-Weber Functions. Technical report Technical Report 53 and AFCRL 796, University Washington Press, Seattle.

ⓘ

Notes:

Reviewed in Math. Comp., v. 17 (1963), pp.315–317.

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Uniform asymptotic smoothing of Stokes’s discontinuities. Proc. Roy. Soc. London Ser. A 422, pp. 7–21.

ⓘ

External Links:

ISSN 0080-4630, FullText, MathReview (André Hautot), ZentralBlatt

Referenced by:

§2.11(iv).

Encodings:

BibTeX

… ►

On a generalization of Barton’s integral and related integrals of complete elliptic integrals. Math. Proc. Cambridge Philos. Soc. 101 (1), pp. 1–5.

ⓘ

External Links:

ISSN 0305-0041, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§19.13(i).

Encodings:

BibTeX

…

36: 10.8 Power Series

… ►When $\nu$ is not an integer the corresponding expansions for $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, and $H_{\nu }^{(2)}\left(z\right)$ are obtained by combining (10.2.2) with (10.2.3), (10.4.7), and (10.4.8). … ►

10.8.1 $$Y_n\left(z\right)=-\frac{{(\frac{1}{2}z)}^{-n}}{\pi }\sum _{k=0}^{n-1}\frac{(n-k-1)!}{k!}{\left(\frac{1}{4}{z}^2\right)}^k+\frac{2}{\pi }\ln \left(\frac{1}{2}z\right)J_n\left(z\right)-\frac{{(\frac{1}{2}z)}^n}{\pi }\sum _{k=0}^{\mathrm{\infty }}(\psi \left(k+1\right)+\psi \left(n+k+1\right))\frac{{(-\frac{1}{4}{z}^2)}^k}{k!(n+k)!},$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

9.1.11

Referenced by:

§10.19(i), §10.7(i)

Permalink:

http://dlmf.nist.gov/10.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.8 and Ch.10

… ►

10.8.2 $$Y_0\left(z\right)=\frac{2}{\pi }\left(\ln \left(\frac{1}{2}z\right)+\gamma \right)J_0\left(z\right)+\frac{2}{\pi }\left(\frac{\frac{1}{4}{z}^2}{{(1!)}^2}-(1+\frac{1}{2})\frac{{(\frac{1}{4}{z}^2)}^2}{{(2!)}^2}+(1+\frac{1}{2}+\frac{1}{3})\frac{{(\frac{1}{4}{z}^2)}^3}{{(3!)}^2}-\mathrm{\cdots }\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

9.1.13

Referenced by:

§10.22(iii), §10.24, §10.7(i)

Permalink:

http://dlmf.nist.gov/10.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.8 and Ch.10

…

37: Bibliography Z

… ►

Integrali di funzioni di Anger, Weber ed Airy-Hardy. Rend. Sem. Mat. Univ. Padova 58, pp. 275–285 (Italian).

ⓘ

External Links:

FullText, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§11.10(x).

Encodings:

BibTeX

►

Su un integrale definito del prodotto di due funzioni di Struve. Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 112 (1-2), pp. 63–81 (Italian).

ⓘ

External Links:

ISSN 0001-4419, MathReview, ZentralBlatt

Referenced by:

§11.7(ii).

Encodings:

BibTeX

… ►

Symmetric elliptic integrals of the third kind. Math. Comp. 24 (109), pp. 199–214.

ⓘ

External Links:

FullText, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.21(i), §19.21(iii), §19.22(iii), §19.25(i), §19.26(i).

Encodings:

BibTeX

…

38: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Often circumstances allow rather stronger statements, such as uniform convergence, or pointwise convergence at points where $f(x)$ is continuous, with convergence to $(f(x_0-)+f(x_0+))/2$ if $x_0$ is an isolated point of discontinuity. … ►where the integral kernel is given by … ►This is the discontinuity across the branch cut in (1.18.52) ${𝝈}_c\subset ℝ$, from $z$ below to above the cut, divided by $2\pi \mathrm{i}$. … ►For generalizations see the Weber transform (10.22.78) and an extended Bessel transform (10.22.79). … ►This representation has poles with residues ${\left|\widehat{f}({\lambda }_n)\right|}^2$ at the discrete eigenvalues and a branch cut along $[0,\mathrm{\infty })$ with discontinuity, from below to above the cut, $2\pi \mathrm{i}{\left|\widehat{f}(\lambda )\right|}^2$, as in (1.18.53), see Newton (2002, §7.1.1). …

39: Software Index

… ► ►►►►►►►

	Open Source													With Book						Commercial
…
6 Exponential, Logarithmic, Sine, and Cosine Integrals
…
11.16(iii) Integrals of …														✓					✓		✓	✓	a
…
11.16(v) ${𝐉}_{\nu }\left(z\right)$, ${𝐄}_{\nu }\left(z\right)$, ${𝐀}_{\nu }\left(z\right)$												a	✓				✓				✓	✓	a
11.16(vi) Integrals of …														✓							✓	✓	a
…
19 Elliptic Integrals
…

…

40: Guide to Searching the DLMF

… ►

Table 1: Query Examples

► ►►►►►►

Query	Matching records contain
…
int sin	the integral of the sin function
int_$^$ sin	any definite integral of sin
…
BesselJ_nu and BesselY_nu	both the Bessel functions $J_{\nu }$ and $Y_{\nu }$.
…
int adj sin	$\int$ immediately followed by $\sin$ without any intervening terms.
…

►

… ►

Table 2: Wildcard Examples

► ►►►

Query	What it stands for
…
int_$^$ sin	any definite integral of sin.

►

…