Jacobi transform

(0.002 seconds)

21—30 of 39 matching pages

21: Bibliography W

… ►

G. N. Watson (1910) The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §15.8(v), §15.8(v).
Encodings:: BibTeX

… ►

F. J. W. Whipple (1927) Some transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 26 (2), pp. 257–272.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

… ►

D. V. Widder (1979) The Airy transform. Amer. Math. Monthly 86 (4), pp. 271–277.

ⓘ

External Links:: ISSN 0002-9890, FullText, Document, MathReview (A. G. Ramm), ZentralBlatt
Referenced by:: (9.10.13), §9.10(ix), §9.10(v).
Encodings:: BibTeX

►

D. V. Widder (1941) The Laplace Transform. Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, NJ.

ⓘ

External Links:: FullText, MathReview (J. D. Tamarkin), ZentralBlatt
Referenced by:: §1.14(vi), §1.15(viii).
Encodings:: BibTeX

… ►

J. Wimp (1964) A class of integral transforms. Proc. Edinburgh Math. Soc. (2) 14, pp. 33–40.

ⓘ

External Links:: Document, MathReview (B. R. Bhonsle), ZentralBlatt
Referenced by:: §13.23(iv).
Encodings:: BibTeX

…

22: Bibliography C

… ►

R. G. Campos (1995) A quadrature formula for the Hankel transform. Numer. Algorithms 9 (2), pp. 343–354.

ⓘ

External Links:: ISSN 1017-1398, Document, MathReview (V. I. Polovinkin), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

►

S. M. Candel (1981) An algorithm for the Fourier-Bessel transform. Comput. Phys. Comm. 23 (4), pp. 343–353.

ⓘ

External Links:: ISSN 0010-4655, Document, MathReview
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

B. C. Carlson (1976) Quadratic transformations of Appell functions. SIAM J. Math. Anal. 7 (2), pp. 291–304.

ⓘ

External Links:: Document, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §16.16(ii), §19.22(i).
Encodings:: BibTeX

… ►

N. B. Christensen (1990) Optimized fast Hankel transform filters. Geophysical Prospecting 38 (5), pp. 545–568.

ⓘ

External Links:: Document
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

P. Cornille (1972) Computation of Hankel transforms. SIAM Rev. 14 (2), pp. 278–285.

ⓘ

External Links:: ISSN 1095-7200, Document, MathReview (R. D. Riess), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

…

23: Bibliography F

… ►

J. Faraut (1982) Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un. J. Funct. Anal. 49 (2), pp. 230–268.

ⓘ

External Links:: ISSN 0022-1236, Document, Link, MathReview (Mogens Flensted-Jensen), ZentralBlatt
Referenced by:: §18.39(i).
Encodings:: BibTeX

… ►

H. E. Fettis (1965) Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.

ⓘ

External Links:: FullText, MathReview (J. E. L. Peck), ZentralBlatt
Referenced by:: §19.36(ii).
Encodings:: BibTeX

… ►

F. Feuillebois (1991) Numerical calculation of singular integrals related to Hankel transform. Comput. Math. Appl. 21 (2-3), pp. 87–94.

ⓘ

Notes:: Fortran, minimum precision 6D.
External Links:: ISSN 0898-1221, Document, MathReview, ZentralBlatt
Referenced by:: §10.77(ix).
Encodings:: BibTeX

… ►

A. S. Fokas and M. J. Ablowitz (1982) On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.13(i), §32.7(i), §32.7(iii).
Encodings:: BibTeX

… ►

C. L. Frenzen (1987b) On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.

ⓘ

External Links:: ISSN 0036-1410, Document, Link, MathReview (Archibald Brown), ZentralBlatt
Referenced by:: §2.3(vi).
Encodings:: BibTeX

…

24: 18.3 Definitions

§18.3 Definitions

►The classical OP’s comprise the Jacobi, Laguerre and Hermite polynomials. … ►For finite power series of the Jacobi, ultraspherical, Laguerre, and Hermite polynomials, see §18.5(iii) (in powers of

x-1

for Jacobi polynomials, in powers of

x

for the other cases). … ►It is also related to a discrete Fourier-cosine transform, see Britanak et al. (2007). … ►

Jacobi on Other Intervals

…

25: 22.17 Moduli Outside the Interval [0,1]

… ►

22.17.1

\operatorname{pq}\left(z,k\right)=\operatorname{pq}\left(z,-k\right),

ⓘ

Symbols:: $\operatorname{pq}\left(\NVar{z},\NVar{k}\right)$ : generic Jacobian elliptic function, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.17.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

… ►

22.17.3

\operatorname{cn}\left(z,1/k\right)=\operatorname{dn}\left(z/k,k\right),

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.11.3
Permalink:: http://dlmf.nist.gov/22.17.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

►

22.17.4

\operatorname{dn}\left(z,1/k\right)=\operatorname{cn}\left(z/k,k\right).

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $z$ : complex and $k$ : modulus
A&S Ref:: 16.11.4
Permalink:: http://dlmf.nist.gov/22.17.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.17(i), §22.17 and Ch.22

… ►

§22.17(ii) Complex Moduli

… ►In particular, the Landen transformations in §§22.7(i) and 22.7(ii) are valid for all complex values of

k

, irrespective of which values of

\sqrt{k}

and

k^{\prime}=\sqrt{1-k^{2}}

are chosen—as long as they are used consistently. …

26: 31.7 Relations to Other Functions

… ►They are analogous to quadratic and cubic hypergeometric transformations (§§15.8(iii)–15.8(v)). … ►Joyce (1994) gives a reduction in which the independent variable is transformed not polynomially or rationally, but algebraically. … ►With

z={\operatorname{sn}}^{2}\left(\zeta,k\right)

and …The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …

27: Errata

… ►We have also incorporated material on continuous

q

-Jacobi polynomials, and several new limit transitions. … ►

Section 1.14

There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.

Transform	New	Abbreviated	Old
	Notation	Notation	Notation
Fourier	$\mathscr{F}\left(f\right)\left(x\right)$	$\mathscr{F}\mskip-3.0muf\mskip 3.0mu\left(x\right)$
Fourier Cosine	$\mathscr{F}_{\mkern-3.0muc}\left(f\right)\left(x\right)$	$\mathscr{F}_{\mkern-3.0muc}\mskip-1.0muf\mskip 3.0mu\left(x\right)$
Fourier Sine	$\mathscr{F}_{\mkern-2.0mus}\left(f\right)\left(x\right)$	$\mathscr{F}_{\mkern-2.0mus}\mskip-1.0muf\mskip 3.0mu\left(x\right)$
Laplace	$\mathscr{L}\left(f\right)\left(s\right)$	$\mathscr{L}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathscr{L}(f(t);s)$
Mellin	$\mathscr{M}\left(f\right)\left(s\right)$	$\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathscr{M}(f;s)$
Hilbert	$\mathcal{H}\left(f\right)\left(s\right)$	$\mathcal{H}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathcal{H}(f;s)$
Stieltjes	$\mathcal{S}\left(f\right)\left(s\right)$	$\mathcal{S}\mskip-3.0muf\mskip 3.0mu\left(s\right)$	$\mathcal{S}(f;s)$

Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.

… ►

Table 22.4.3

Originally a minus sign was missing in the entries for $\operatorname{cd}u$ and $\operatorname{dc}u$ in the second column (headed $z+K+iK^{\prime}$ ). The correct entries are $-k^{-1}\operatorname{ns}z$ and $-k\operatorname{sn}z$ . Note: These entries appear online but not in the published print edition. More specifically, Table 22.4.3 in the published print edition is restricted to the three Jacobian elliptic functions $\operatorname{sn},\operatorname{cn},\operatorname{dn}$ , whereas Table 22.4.3 covers all 12 Jacobian elliptic functions.

	$u$
	$z+K$	$z+K+\mathrm{i}K^{\prime}$	$z+\mathrm{i}K^{\prime}$	$z+2K$	$z+2K+2\mathrm{i}K^{\prime}$	$z+2\mathrm{i}K^{\prime}$
$\vdots$
$\operatorname{cd}u$	$-\operatorname{sn}z$	$-k^{-1}\operatorname{ns}z$	$k^{-1}\operatorname{dc}z$	$-\operatorname{cd}z$	$-\operatorname{cd}z$	$\operatorname{cd}z$
$\vdots$
$\operatorname{dc}u$	$-\operatorname{ns}z$	$-k\operatorname{sn}z$	$k\operatorname{cd}z$	$-\operatorname{dc}z$	$-\operatorname{dc}z$	$\operatorname{dc}z$
$\vdots$

Reported 2014-02-28 by Svante Janson.

… ►

Equation (22.6.7)

22.6.7

\operatorname{dn}\left(2z,k\right)=\frac{{\operatorname{dn}}^{2}\left(z,k% \right)-k^{2}{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{cn}}^{2}% \left(z,k\right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)}=\frac{{% \operatorname{dn}}^{4}\left(z,k\right)+k^{2}{k^{\prime}}^{2}{\operatorname{sn}% }^{4}\left(z,k\right)}{1-k^{2}{\operatorname{sn}}^{4}\left(z,k\right)}

Originally the term $k^{2}{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{cn}}^{2}\left(z,k\right)$ was given incorrectly as $k^{2}{\operatorname{sn}}^{2}\left(z,k\right){\operatorname{dn}}^{2}\left(z,k\right)$ .

Reported 2014-02-28 by Svante Janson.

… ►

Table 22.5.4

Originally the limiting form for $\operatorname{sc}\left(z,k\right)$ in the last line of this table was incorrect ( $\cosh z$ , instead of $\sinh z$ ).

$\operatorname{sn}\left(z,k\right)$ $\;\to\;$	$\tanh z$	$\operatorname{cd}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{dc}\left(z,k\right)$ $\;\to\;$	$1$	$\operatorname{ns}\left(z,k\right)$ $\;\to\;$	$\coth z$
$\operatorname{cn}\left(z,k\right)$ $\;\to\;$	$\operatorname{sech}z$	$\operatorname{sd}\left(z,k\right)$ $\;\to\;$	$\sinh z$	$\operatorname{nc}\left(z,k\right)$ $\;\to\;$	$\cosh z$	$\operatorname{ds}\left(z,k\right)$ $\;\to\;$	$\operatorname{csch}z$
$\operatorname{dn}\left(z,k\right)$ $\;\to\;$	$\operatorname{sech}z$	$\operatorname{nd}\left(z,k\right)$ $\;\to\;$	$\cosh z$	$\operatorname{sc}\left(z,k\right)$ $\;\to\;$	$\sinh z$	$\operatorname{cs}\left(z,k\right)$ $\;\to\;$	$\operatorname{csch}z$

Reported 2010-11-23.

…

28: 22.15 Inverse Functions

… ►are denoted respectively by ►

\xi=\operatorname{arcsn}\left(x,k\right),

►

\eta=\operatorname{arccn}\left(x,k\right),

… ►Equations (22.15.1) and (22.15.4), for

\operatorname{arcsn}\left(x,k\right)

, are equivalent to (22.15.12) and also to … ►can be transformed into normal form by elementary change of variables. …

29: 18.25 Wilson Class: Definitions

… ►For the Wilson class OP’s

p_{n}(x)

with

x=\lambda(y)

: if the

y

-orthogonality set is

\{0,1,\dots,N\}

, then the role of the differentiation operator

\ifrac{\mathrm{d}}{\mathrm{d}x}

in the Jacobi, Laguerre, and Hermite cases is played by the operator

\Delta_{y}

followed by division by

\Delta_{y}(\lambda(y))

, or by the operator

\nabla_{y}

followed by division by

\nabla_{y}(\lambda(y))

. … ►Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials

W_{n}\left(x;a,b,c,d\right)

, continuous dual Hahn polynomials

S_{n}\left(x;a,b,c\right)

, Racah polynomials

R_{n}\left(x;\alpha,\beta,\gamma,\delta\right)

, and dual Hahn polynomials

R_{n}\left(x;\gamma,\delta,N\right)

. ►

Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.

► ►►

OP	$p_{n}(x)$	$x=\lambda(y)$	Orthogonality range for $y$	Constraints
…

► …

30: Bibliography H

… ►

P. I. Hadži (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

ⓘ

External Links:: MathReview (L. Berg)
Referenced by:: §7.14(iii).
Encodings:: BibTeX

… ►

E. Hahn (1980) Asymptotik bei Jacobi-Polynomen und Jacobi-Funktionen. Math. Z. 171 (3), pp. 201–226 (German).

ⓘ

External Links:: ISSN 0025-5874, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §18.15(i).
Encodings:: BibTeX

… ►

N. Hale and A. Townsend (2016) A fast FFT-based discrete Legendre transform. IMA J. Numer. Anal. 36 (4), pp. 1670–1684.

ⓘ

External Links:: ISSN 0272-4979, Document, Link, MathReview (Denis N. Sidorov)
Referenced by:: §18.16(iii).
Encodings:: BibTeX

… ►

R. A. Handelsman and J. S. Lew (1970) Asymptotic expansion of Laplace transforms near the origin. SIAM J. Math. Anal. 1 (1), pp. 118–130.

ⓘ

External Links:: Document, MathReview (E. Riekstiņš), ZentralBlatt
Referenced by:: §2.5(iii), §2.5(ii).
Encodings:: BibTeX

… ►

E. W. Hansen (1985) Fast Hankel transform algorithm. IEEE Trans. Acoust. Speech Signal Process. 32 (3), pp. 666–671.

ⓘ

External Links:: ISSN 0096-3518, FullText, ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

…