Jacobi transform

SearchAdvancedHelp

(0.002 seconds)

1—10 of 47 matching pages

1: 14.31 Other Applications

… ►

§14.31(ii) Conical Functions

… ►The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …

2: 15.17 Mathematical Applications

… ►Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. …

3: 15.9 Relations to Other Functions

… ►The Jacobi transform is defined as …with inverse … …

4: 22.6 Elementary Identities

… ►

§22.6(iv) Rotation of Argument (Jacobi’s Imaginary Transformation)

►

Table 22.6.1: Jacobi’s imaginary transformation of Jacobian elliptic functions.

► ►►

$\mathrm{sn}(\mathrm{i}z,k)$ $=$	$\mathrm{i}\mathrm{sc}(z,k^{\prime })$	$\mathrm{dc}(\mathrm{i}z,k)$ $=$	$\mathrm{dn}(z,k^{\prime })$
…

►

…

5: Bibliography K

… ►

A new proof of a Paley-Wiener type theorem for the Jacobi transform. Ark. Mat. 13, pp. 145–159.

ⓘ

External Links:

Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

BibTeX

…

6: 1.14 Integral Transforms

§1.14 Integral Transforms

…

7: 18.17 Integrals

… ►

Jacobi

… ►

Jacobi

… ►

Jacobi

►

18.17.36 $${\int }_{-1}^1{(1-x)}^{z-1}{(1+x)}^{\beta }{P}_n^{(\alpha ,\beta )}\left(x\right)dx=\frac{2^{\beta +z}\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(1+\beta +n\right){\left(1+\alpha -z\right)}_n}{n!\mathrm{\Gamma }\left(1+\beta +z+n\right)},$$ $\mathrm{\Re }z>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

…

8: 37.4 Disk with Weight Function ${\left(1-x^2-y^2\right)}^{\alpha }$

… ►

$$ℱ\left(f\right)\left(\rho \cos \theta ,\rho \sin \theta \right)={\mathrm{i}}^{m-n}{e}_{m-n}(\theta ){\int }_0^1{r}^{m-n+1}{(1-r^2)}^{\alpha }{R}_n^{(\alpha ,m-n)}\left(2r^2-1\right)J_{m-n}\left(\rho r\right)dr=2^{\alpha }\mathrm{\Gamma }\left(\alpha +1\right){\mathrm{i}}^{m+n}{e}_{m-n}(\theta ){\rho }^{-(\alpha +1)}{J}_{m+n+\alpha +1}\left(\rho \right),$$ $\rho >0$,

… ►First, the Jacobi polynomials (37.3.3) on $\mathrm{△}$ and the ultraspherical polynomials (37.4.4) on $𝔻$ are related by the quadratic transformations …Second, the Jacobi polynomials (37.3.9) on $\mathrm{△}$ are related to the real disk polynomials (37.4.15) by the quadratic transformations …

9: 37.12 Orthogonal Polynomials on Quadratic Surfaces

… ►

$S_{\mathrm{\ell },m}^n(z^2,{\left|z\right|}^2;0)=\left\{\begin{array}{c}{z}^{-1}{S}_{n+m+1,n-m}(z,\overline{z})\\ {\overline{z}}^{-1}{S}_{n-m,n+m+1}(z,\overline{z})\end{array}\right\}$ , $0\le m\le n$, $z\in ℂ$.

10: 29.10 Lamé Functions with Imaginary Periods

… ►transform (29.2.1) into ►

29.10.3 $$\frac{d^{2}w}{{d{z}^{\prime }}^2}+(h^{\prime }-\nu (\nu +1)k_{}^{\prime }{}_{}{}^2{\mathrm{sn}}^2(z^{\prime },k^{\prime }))w=0.$$

ⓘ

Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $h$: real parameter, $k$: real parameter, $\nu$: real parameter and $w(z)$: solution

Permalink:

http://dlmf.nist.gov/29.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §29.10 and Ch.29

…