# SL(2,Z) bilinear transformation

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##### 11: 25.3 Graphics Figure 25.3.2: Riemann zeta function ζ ⁡ ( x ) and its derivative ζ ′ ⁡ ( x ) , - 12 ≤ x ≤ - 2 . Magnify Figure 25.3.4: Z ⁡ ( t ) , 0 ≤ t ≤ 50 . Z ⁡ ( t ) and ζ ⁡ ( 1 2 + i ⁢ t ) have the same zeros. … Magnify Figure 25.3.5: Z ⁡ ( t ) , 1000 ≤ t ≤ 1050 . Magnify Figure 25.3.6: Z ⁡ ( t ) , 10000 ≤ t ≤ 10050 . 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. …
##### 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
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
###### BilinearTransformation
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 $R_{-a}\left(\boldsymbol{{\tfrac{1}{2}}};\mathbf{z}\right)=A^{-a}\sum_{N=0}^{% \infty}\frac{{\left(a\right)_{N}}}{{\left(\tfrac{1}{2}n\right)_{N}}}T_{N}(% \boldsymbol{{\tfrac{1}{2}}},\mathbf{Z}),$
##### 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 transformFor 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.
• 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.
• 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.
• 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.
• F. H. L. Essler, H. Frahm, A. R. Its, and V. E. Korepin (1996) Painlevé transcendent describes quantum correlation function of the $XXZ$ antiferromagnet away from the free-fermion point. J. Phys. A 29 (17), pp. 5619–5626.
• ##### 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.
• N. I. Akhiezer (1988) Lectures on Integral Transforms. Translations of Mathematical Monographs, Vol. 70, American Mathematical Society, Providence, RI.
• 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.
• G. E. Andrews (1972) Summations and transformations for basic Appell series. J. London Math. Soc. (2) 4, pp. 618–622.
• 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.