Jacobi matrix

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11: 21.2 Definitions

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21.2.1 $$\theta \left(𝐳\right|𝛀)=\sum _{𝐧\in {\mathbb{Z}}^g}{\mathrm{e}}^{2\pi \mathrm{i}\left(\frac{1}{2}𝐧\cdot 𝛀\cdot 𝐧+𝐧\cdot 𝐳\right)}.$$

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Defines:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $g$: positive integer and $𝛀$: a Riemann matrix

Referenced by:

§21.2(ii)

Permalink:

http://dlmf.nist.gov/21.2.E1

Encodings:

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See also:

Annotations for §21.2(i), §21.2 and Ch.21

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21.2.7 $$\theta \left[\genfrac{}{}{0.0pt}{}{\mathrm{𝟎}}{\mathrm{𝟎}}\right]\left(𝐳\right|𝛀)=\theta \left(𝐳\right|𝛀).$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(ii), §21.2 and Ch.21

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21.2.8 $$\theta \left(z\right|\mathrm{\Omega })={\theta }_3\left(\pi z\right|\mathrm{\Omega }),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left(𝐳\right|𝛀)$: Riemann theta function and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/21.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(iii), §21.2 and Ch.21

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21.2.9 $${\theta }_1\left(\pi z\right|\mathrm{\Omega })=-\theta \left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}}{\frac{1}{2}}\right]\left(z\right|\mathrm{\Omega }),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/21.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(iii), §21.2 and Ch.21

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21.2.10 $${\theta }_2\left(\pi z\right|\mathrm{\Omega })=\theta \left[\genfrac{}{}{0.0pt}{}{\frac{1}{2}}{0}\right]\left(z\right|\mathrm{\Omega }),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics and $\pi$: the ratio of the circumference of a circle to its diameter

Permalink:

http://dlmf.nist.gov/21.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §21.2(iii), §21.2 and Ch.21

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12: Bibliography D

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Detection of the density matrix through optical homodyne tomography without filtered back projection. Phys. Rev. A 50 (5), pp. 4298–4302.

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External Links:

Document

Referenced by:

§13.28(iii).

Encodings:

BibTeX

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Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 (11), pp. 1335–1425.

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External Links:

ISSN 0010-3640, Document, MathReview (D. S. Lubinsky), ZentralBlatt

Referenced by:

§18.32.

Encodings:

BibTeX

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Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions. Methods Appl. Anal. 6 (3), pp. 21–56.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§15.12(iii), §18.15(i), §18.15(vi).

Encodings:

BibTeX

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A survey on orthogonal matrix polynomials satisfying second order differential equations. J. Comput. Appl. Math. 178 (1-2), pp. 169–190.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§18.36(iv).

Encodings:

BibTeX

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Product formulas and Nicholson-type integrals for Jacobi functions. I. Summary of results. SIAM J. Math. Anal. 9 (1), pp. 76–86.

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External Links:

Document, MathReview (R. C. Varma), ZentralBlatt

Referenced by:

§18.17(iii).

Encodings:

BibTeX

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13: 18.36 Miscellaneous Polynomials

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§18.36(i) Jacobi-Type Polynomials

►These are OP’s on the interval $(-1,1)$ with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at $-1$ and $1$ to the weight function for the Jacobi polynomials. … ►

§18.36(iv) Orthogonal Matrix Polynomials

►These are matrix-valued polynomials that are orthogonal with respect to a square matrix of measures on the real line. Classes of such polynomials have been found that generalize the classical OP’s in the sense that they satisfy second order matrix differential equations with coefficients independent of the degree. …

14: Bibliography H

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

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External Links:

ISSN 0025-5874, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.15(i).

Encodings:

BibTeX

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The combination of $R$-matrix and complex coordinate methods: Application to the diamagnetic Rydberg spectra of Ba and Sr. J. Phys. B 26 (12), pp. 1775–1790.

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External Links:

ISSN 0953-4075, Document, MathReview

Referenced by:

4th item.

Encodings:

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On an “equivalent quadrature” calculation of matrix elements of ${(z-p^2/2m)}^{-1}$ using an $L^2$ expansion technique. J. Comput. Phys. 13, pp. 536–550.

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External Links:

Document, Link

Referenced by:

§18.39(iv).

Encodings:

BibTeX

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Bessel functions of matrix argument. Ann. of Math. (2) 61 (3), pp. 474–523.

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External Links:

FullText, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§35.1, §35.2, §35.3(ii), §35.5(i), §35.5(ii), §35.5(iii), §35.6(i), §35.6(ii), §35.6(iii), §35.6(iv), §35.7(i), §35.7(ii), §35.7(iv), §35.8(i), §35.8(ii), §35.8(iv), Ch.35, Notations P.

Encodings:

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15: Bibliography J

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Fundamenta Nova Theoriae Functionum Ellipticarum. Regiomonti, Sumptibus fratrum Bornträger.

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Notes:

Reprinted in Werke, Vol. I, pp. 49–239, Herausgegeben von C. W. Borchardt, Berlin, Verlag von G. Reimer, 1881.

External Links:

Werke TOC

Referenced by:

§27.13(iv).

Encodings:

BibTeX

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Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35 (2), pp. 475–501.

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External Links:

FullText, MathReview (I. Olkin), ZentralBlatt

Referenced by:

§35.11, §35.4(i), §35.4(ii), §35.8(i), §35.8(ii), §35.8(iv), §35.9, Ch.35.

Encodings:

BibTeX

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Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent. Phys. D 1 (1), pp. 80–158.

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External Links:

ISSN 0167-2789, Document, MathReview, ZentralBlatt

Referenced by:

§32.16.

Encodings:

BibTeX

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