SL(2,Z) bilinear transformation

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21: 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

… ►

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, (18.17.21_1), (18.17.34_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

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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

… ►

W. N. Everitt (1982) On the transformation theory of ordinary second-order linear symmetric differential expressions. Czechoslovak Math. J. 32(107) (2), pp. 275–306.

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Notes:: With a loose Russian summary
External Links:: ISSN 0011-4642, MathReview (D. Hinton)
Referenced by:: §1.13(viii).
Encodings:: BibTeX

…

22: 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 $z$ .

…

23: 15.14 Integrals

§15.14 Integrals

►The Mellin transform of the hypergeometric function of negative argument is given by … ►Fourier transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §§1.14 and 2.14). Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §4.21), Oberhettinger and Badii (1973, §1.19), and Prudnikov et al. (1992a, §3.37). …Hankel transforms of hypergeometric functions are given in Oberhettinger (1972, §1.17) and Erdélyi et al. (1954b, §8.17). …

24: Bibliography P

… ►

R. B. Paris (2005a) A Kummer-type transformation for a

{}_{2}F_{2}

hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

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External Links:: ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

… ►

R. Parnes (1972) Complex zeros of the modified Bessel function

K_{n}(Z)

. Math. Comp. 26 (120), pp. 949–953.

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External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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E. Petropoulou (2000) Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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External Links:: ISSN 1065-2469, Document, MathReview, ZentralBlatt
Referenced by:: §10.37.
Encodings:: BibTeX

… ►

A. Pinkus and S. Zafrany (1997) Fourier Series and Integral Transforms. Cambridge University Press, Cambridge.

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External Links:: ISBN 0-521-59209-7; 0-521-59771-4, MathReview, ZentralBlatt
Referenced by:: §1.14(vii).
Encodings:: BibTeX

… ►

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev (1992a) Integrals and Series: Direct Laplace Transforms, Vol. 4. Gordon and Breach Science Publishers, New York.

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Notes:: Table erratum: Math. Comp. v. 66 (1997), no. 220, p. 1766.
External Links:: ISBN 2-88124-837-3, MathReview (J. M. H. Peters), ZentralBlatt
Referenced by:: §1.14(viii), §1.4(iv), §10.22(vi), §10.43(vi), §10.71(iii), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(ii), §13.10(vi), §13.23(i), §14.17(v), §15.14, §16.20, §16.5, §18.17(ix), §19.13(iii), §20.10(iii), §24.13(iii), §25.11(ix), §25.12(ii), §25.7, §5.13, §6.14(iii), §7.14(iii), §8.14, §9.10(ix).
Encodings:: BibTeX

…

25: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►

Kummer Transformation

►Let

c=b_{1}+b_{2}-a_{1}-a_{2}-a_{3}

. … ►

Thomae Transformation

►Again, let

c=b_{1}+b_{2}-a_{1}-a_{2}-a_{3}

. … ►

Laplace Transform

…

26: 2.5 Mellin Transform Methods

… ► … ►In the half-plane

\Re z>\max(0,-2\nu)

, the product

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(1-z\right)\mathscr{M}\mskip-3.0muh% \mskip 3.0mu\left(z\right)

has a pole of order two at each positive integer, and … ►By Table 2.5.1,

\mathscr{M}\mskip-3.0muf_{2}\mskip 3.0mu\left(z\right)

is an analytic function in the half-plane

\Re z<b

. … ►Alternatively, if

\kappa\neq 0

in (2.5.18), then

\mathscr{M}\mskip-3.0muh_{2}\mskip 3.0mu\left(z\right)

can be continued analytically to an entire function. ►Since

\mathscr{M}\mskip-3.0muh_{1}\mskip 3.0mu\left(z\right)

is analytic for

\Re z>-c

by Table 2.5.1, the analytically-continued

\mathscr{M}\mskip-3.0muh_{2}\mskip 3.0mu\left(z\right)

allows us to extend the Mellin transform of

h

via …

27: 22.7 Landen Transformations

§22.7 Landen Transformations

►

§22.7(i) Descending Landen Transformation

… ►

§22.7(ii) Ascending Landen Transformation

… ►

k_{2}=\frac{2\sqrt{k}}{1+k},

… ►

§22.7(iii) Generalized Landen Transformations

…

28: 22.16 Related Functions

… ►With

q

as in (22.2.1) and

\zeta=\pi x/(2K)

, … ►In Equations (22.16.24)–(22.16.26),

-2K<x<2K

. … ►where

\xi=x/{\theta_{3}}^{2}\left(0,q\right)

. … ►(Sometimes in the literature

\mathrm{Z}\left(x|k\right)

is denoted by

\mathrm{Z}(\operatorname{am}\left(x,k\right),k^{2})

.) … ►

\mathrm{Z}\left(x|k\right)

satisfies the same quasi-addition formula as the function

\mathcal{E}\left(x,k\right)

, given by (22.16.27). …

29: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►

35.7.1

{{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}% {k!}\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}{\left[b\right]_{\kappa}}}% {{\left[c\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right)},

-c+\frac{1}{2}(j+1)\notin\mathbb{N}

1\leq j\leq m

;

\|\mathbf{T}\|<1

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $!$ : factorial (as in $n!$ ), $\notin$ : not an element of, $\mathbb{N}$ : set of all positive integers, ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $\|\mathbf{T}\|$ : spectral norm of $\mathbf{T}$ , $c$ : complex variable, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : positive integer and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(i), §35.7 and Ch.35

… ►

Case $m=2$

►

35.7.3

{{}_{2}F_{1}}\left({a,b\atop c};\begin{bmatrix}t_{1}&0\\ 0&t_{2}\end{bmatrix}\right)=\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{% \left(c-a\right)_{k}}{\left(b\right)_{k}}{\left(c-b\right)_{k}}}{k!\,{\left(c% \right)_{2k}}{\left(c-\tfrac{1}{2}\right)_{k}}}\*(t_{1}t_{2})^{k}{{}_{2}F_{1}}% \left({a+k,b+k\atop c+2k};t_{1}+t_{2}-t_{1}t_{2}\right).

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $a$ : complex variable, $t_{j}$ : eigenvalues of $\mathbf{T}$ , $b$ : complex variable, $c$ : complex variable and $k$ : nonnegative integer
Referenced by:: Erratum (V1.0.25) for Equation (35.7.3)
Permalink:: http://dlmf.nist.gov/35.7.E3
Encodings:: pMML, png, TeX
Notational change (effective with 1.0.25):: Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${{}_{2}F_{1}}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

… ►

Transformations of Parameters

… ►Subject to the conditions (a)–(c), the function

f(\mathbf{T})={{}_{2}F_{1}}\left(a,b;c;\mathbf{T}\right)

is the unique solution of each partial differential equation …

30: 19.15 Advantages of Symmetry

… ►The function

R_{-a}\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_{n}\right)

(Carlson (1963)) reveals the full permutation symmetry that is partially hidden in

F_{D}

, and leads to symmetric standard integrals that simplify many aspects of theory, applications, and numerical computation. ►Symmetry in

x, y, z

R_{F}\left(x,y,z\right)

R_{G}\left(x,y,z\right)

, and

R_{J}\left(x,y,z,p\right)

replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9). (19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. …