Jacobi transform

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1: 14.31 Other Applications

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§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

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

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

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$\mathrm{sn}(\mathrm{i}z,k)$ $=$	$\mathrm{i}\mathrm{sc}(z,k^{\prime })$	$\mathrm{dc}(\mathrm{i}z,k)$ $=$	$\mathrm{dn}(z,k^{\prime })$
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5: Bibliography K

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

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External Links:

Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

§18.10(i).

Encodings:

BibTeX

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6: 18.17 Integrals

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Jacobi

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Jacobi

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Jacobi

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

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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

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7: 1.14 Integral Transforms

§1.14 Integral Transforms

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8: 29.10 Lamé Functions with Imaginary Periods

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

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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

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9: 20.7 Identities

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20.7.33 $${(-\mathrm{i}\tau )}^{1/2}{\theta }_4\left(z\right|\tau )=\exp \left(\mathrm{i}{\tau }^{\prime }{z}^2/\pi \right){\theta }_2\left(z{\tau }^{\prime }\right|{\tau }^{\prime }).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $z$: complex and $\tau$: lattice parameter

Referenced by:

§20.13

Permalink:

http://dlmf.nist.gov/20.7.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §20.7(viii), §20.7 and Ch.20

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10: 20.10 Integrals

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20.10.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}{\theta }_2\left(0\right|\mathrm{i}{x}^2)dx=2^s(1-2^{-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit and $\int$: integral

Keywords:

Mellin transform

Referenced by:

§20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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20.10.2 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}({\theta }_3\left(0\right|\mathrm{i}{x}^2)-1)dx={\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit and $\int$: integral

Keywords:

Mellin transform

Referenced by:

§20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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20.10.3 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}(1-{\theta }_4\left(0\right|\mathrm{i}{x}^2))dx=(1-2^{1-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit and $\int$: integral

Keywords:

Mellin transform

Referenced by:

§20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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