SL(2,Z) bilinear transformation

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11: 25.3 Graphics

… ►

See accompanying text — Figure 25.3.2: Riemann zeta function $\zeta\left(x\right)$ and its derivative $\zeta'\left(x\right)$ , $-12\leq x\leq-2$ . Magnify

… ►

►

12: 18.38 Mathematical Applications

… ►The scaled Chebyshev polynomial

2^{1-n}T_{n}\left(x\right)

n\geq 1

, enjoys the “minimax” property on the interval

[-1,1]

, that is,

|2^{1-n}T_{n}\left(x\right)|

has the least maximum value among all monic polynomials of degree

n

. … ►

Integrable Systems

… ►It has elegant structures, including

N

-soliton solutions, Lax pairs, and Bäcklund transformations. …However, by using Hirota’s technique of bilinear formalism of soliton theory, Nakamura (1996) shows that a wide class of exact solutions of the Toda equation can be expressed in terms of various special functions, and in particular classical OP’s. … ►

Radon Transform

…

13: 25.10 Zeros

… ►Calculations relating to the zeros on the critical line make use of the real-valued function …is chosen to make

Z(t)

real, and

\operatorname{ph}\Gamma\left(\frac{1}{4}+\frac{1}{2}it\right)

assumes its principal value. Because

|Z(t)|=|\zeta\left(\frac{1}{2}+it\right)|

Z(t)

vanishes at the zeros of

\zeta\left(\frac{1}{2}+it\right)

, which can be separated by observing sign changes of

Z(t)

. Because

Z(t)

changes sign infinitely often,

\zeta\left(\frac{1}{2}+it\right)

has infinitely many zeros with

t

real. … ►Sign changes of

Z(t)

are determined by multiplying (25.9.3) by

\exp\left(i\vartheta(t)\right)

to obtain the Riemann–Siegel formula: …

14: 19.36 Methods of Computation

… ►

§19.36(ii) Quadratic Transformations

… ►Descending Gauss transformations of

\Pi\left(\phi,\alpha^{2},k\right)

(see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). …If

\alpha^{2}=k^{2}

, then the method fails, but the function can be expressed by (19.6.13) in terms of

E\left(\phi,k\right)

, for which Neuman (1969b) uses ascending Landen transformations. … ►Quadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)). …The function

\mathrm{el2}\left(x,k_{c},a,b\right)

is computed by descending Landen transformations if

x

is real, or by descending Gauss transformations if

x

is complex (Bulirsch (1965b)). …

15: 35.4 Partitions and Zonal Polynomials

… ►For any partition

\kappa

, the zonal polynomial

Z_{\kappa}:\boldsymbol{\mathcal{S}}\to\mathbb{R}

is defined by the properties ►

35.4.2

Z_{\kappa}\left(\mathbf{I}\right)=|\kappa|!\,2^{2|\kappa|}\,{\left[m/2\right]_% {\kappa}}\frac{\prod\limits_{1\leq j<l\leq\ell(\kappa)}(2k_{j}-2k_{l}-j+l)}{% \prod\limits_{j=1}^{\ell(\kappa)}(2k_{j}+\ell(\kappa)-j)!}

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Symbols:: $!$ : factorial (as in $n!$ ), ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\mathbf{I}$ : $m\times m$ identity matrix, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : positive integer, $\kappa$ : vector of nonnegative integers and $\ell(\kappa)$ : number of nonzeros
Referenced by:: §35.4(i)
Permalink:: http://dlmf.nist.gov/35.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §35.4(i), §35.4 and Ch.35

… ►Therefore

Z_{\kappa}\left(\mathbf{T}\right)

is a symmetric polynomial in the eigenvalues of

\mathbf{T}

. … ►For

k=0,1,2,\dots

, … ►For

\mathbf{T}\in{\boldsymbol{\Omega}}

and

\Re\left(a\right),\Re\left(b\right)>\frac{1}{2}(m-1)

, …

16: 1.9 Calculus of a Complex Variable

… ► ►

Bilinear Transformation

… ►The cross ratio of

z_{1},z_{2},z_{3},z_{4}\in\mathbb{C}\cup\{\infty\}

is defined by …or its limiting form, and is invariant under bilinear transformations. ►Other names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation. …

17: 19.19 Taylor and Related Series

… ►For

N=0,1,2,\dots

define the homogeneous hypergeometric polynomial … ►If

n=2

, then (19.19.3) is a Gauss hypergeometric series (see (19.25.43) and (15.2.1)). … ►and define the

n

-tuple

\mathbf{\tfrac{1}{2}}=(\tfrac{1}{2},\dots,\tfrac{1}{2})

. … ►The number of terms in

T_{N}

can be greatly reduced by using variables

\mathbf{Z}=\boldsymbol{{1}}-(\mathbf{z}/A)

with

A

chosen to make

E_{1}(\mathbf{Z})=0

. … ►

19.19.7

R_{-a}\left(\boldsymbol{{\tfrac{1}{2}}};\mathbf{z}\right)=A^{-a}\sum_{N=0}^{% \infty}\frac{{\left(a\right)_{N}}}{{\left(\tfrac{1}{2}n\right)_{N}}}T_{N}(% \boldsymbol{{\tfrac{1}{2}}},\mathbf{Z}),

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Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $n$ : nonnegative integer, $N$ : nonnegative integer, $T_{N}(\mathbf{b},\mathbf{z})$ : polynomial, $\mathbf{Z}$ : variable and $A$
Referenced by:: §19.15, §19.36(i), §19.36(i), §19.36(i)
Permalink:: http://dlmf.nist.gov/19.19.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.19 and Ch.19

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18: 2.4 Contour Integrals

… ►For examples and extensions (including uniformity and loop integrals) see Olver (1997b, Chapter 4), Wong (1989, Chapter 1), and Temme (1985). ►

§2.4(ii) Inverse Laplace Transforms

… ►Then the Laplace transform … ►For examples see Olver (1997b, pp. 315–320). … ►

(b)

$z$ ranges along a ray or over an annular sector $\theta_{1}\leq\theta\leq\theta_{2}$ , $|z|\geq Z$ , where $\theta=\operatorname{ph}z$ , $\theta_{2}-\theta_{1}<\pi$ , and $Z>0$ . $I(z)$ converges at $b$ absolutely and uniformly with respect to $z$ .

…

19: Bibliography E

… ►

U. T. Ehrenmark (1995) The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature. J. Comput. Appl. Math. 61 (1), pp. 43–72.

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External Links:: ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

Á. Elbert and A. Laforgia (1994) Interlacing properties of the zeros of Bessel functions. Atti Sem. Mat. Fis. Univ. Modena XLII (2), pp. 525–529.

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External Links:: ISSN 0041-8986, MathReview (Evangelos Ifantis), ZentralBlatt
Referenced by:: §10.21(i).
Encodings:: BibTeX

… ►

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi (1954a) Tables of Integral Transforms. Vol. I. McGraw-Hill Book Company, Inc., New York-Toronto-London.

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Notes:: Table errata: Math. Comp. v. 66 (1997), no. 220, p. 1766–1767, v. 65 (1996), no. 215, p. 1384, v. 50 (1988), no. 182, p. 653, v. 41 (1983), no. 164, p. 778–779, v. 27 (1973), no. 122, p. 451, v. 26 (1972), no. 118, p. 599, v. 25 (1971), no. 113, p. 199, v. 24 (1970), no. 109, p. 239-240.
External Links:: MathReview (H. Kober), ZentralBlatt
Referenced by:: §1.14(viii), §10.22(vi), §10.32(iv), §10.43(vi), §10.9(iv), §11.10(x), §11.5(iii), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(ii), §13.10(iii), §13.10(iv), §13.23(i), §13.23(ii), §14.17(v), §15.14, §16.20, §16.5, §18.17(ix), §20.10(ii), §20.10(iii), §25.5(ii), §5.13, §6.14(iii), §6.7(iv), §7.14(iii), §7.7(iii), §8.14.
Encodings:: BibTeX

►

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi (1954b) Tables of Integral Transforms. Vol. II. McGraw-Hill Book Company, Inc., New York-Toronto-London.

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Notes:: Table errata: Math. Comp. v. 65 (1996), no. 215, p. 1385, v. 41 (1983), no. 164, pp. 779–780, v. 31 (1977), no. 138, p. 614, v. 31 (1977), no. 137, pp. 328–329, v. 26 (1972), no. 118, p. 599, v. 25 (1971), no. 113, p. 199, v. 23 (1969), no. 106, p. 468.
External Links:: MathReview (H. Kober), ZentralBlatt
Referenced by:: §1.14(viii), §10.22(v), §10.22(vi), §10.43(v), §10.43(vi), §12.12, §13.10(v), §13.10(v), §13.10(vi), §13.23(iii), §13.23(v), §15.14, §15.14, §18.17(ix), §5.13, §8.14.
Encodings:: BibTeX

… ►

F. H. L. Essler, H. Frahm, A. R. Its, and V. E. Korepin (1996) Painlevé transcendent describes quantum correlation function of the

X X Z

antiferromagnet away from the free-fermion point. J. Phys. A 29 (17), pp. 5619–5626.

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External Links:: ISSN 0305-4470, Document, MathReview (Estelle L. Basor), ZentralBlatt
Referenced by:: §32.16.
Encodings:: BibTeX

…

20: Bibliography

… ►

M. J. Ablowitz and H. Segur (1981) Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics, Vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:: ISBN 0-89871-174-6, MathReview (Benno Fuchssteiner), ZentralBlatt
Referenced by:: §21.9, §21.9, §32.5, §9.16, Profile Mark J. Ablowitz.
Encodings:: BibTeX

… ►

N. I. Akhiezer (1988) Lectures on Integral Transforms. Translations of Mathematical Monographs, Vol. 70, American Mathematical Society, Providence, RI.

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Notes:: Translated from the Russian by H. H. McFaden
External Links:: ISBN 0-8218-4524-1, MathReview, ZentralBlatt
Referenced by:: §10.22(v).
Encodings:: BibTeX

… ►

F. V. Andreev and A. V. Kitaev (2002) Transformations

RS^{2}_{4}(3)

of the ranks

\leq 4

and algebraic solutions of the sixth Painlevé equation. Comm. Math. Phys. 228 (1), pp. 151–176.

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External Links:: ISSN 0010-3616, Document, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.9(iii).
Encodings:: BibTeX

… ►

G. E. Andrews (1972) Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.

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External Links:: MathReview (R. N. Jain), ZentralBlatt
Referenced by:: §17.11.
Encodings:: BibTeX

… ►

R. Askey (1974) Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum. SIAM J. Math. Anal. 5, pp. 119–124.

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External Links:: ISSN 1095-7154, Document, MathReview (R. P. Soni), ZentralBlatt
Referenced by:: §18.18(iv).
Encodings:: BibTeX

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