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SL(2,Z) bilinear transformation

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11: 25.3 Graphics
See accompanying text
Figure 25.3.2: Riemann zeta function ζ ( x ) and its derivative ζ ( x ) , - 12 x - 2 . Magnify
See accompanying text
Figure 25.3.4: Z ( t ) , 0 t 50 . Z ( t ) and ζ ( 1 2 + i t ) have the same zeros. … Magnify
See accompanying text
Figure 25.3.5: Z ( t ) , 1000 t 1050 . Magnify
See accompanying text
Figure 25.3.6: Z ( t ) , 10000 t 10050 . Magnify
12: 18.38 Mathematical Applications
The scaled Chebyshev polynomial 2 1 - n T n ( x ) , n 1 , enjoys the “minimax” property on the interval [ - 1 , 1 ] , that is, | 2 1 - n T n ( x ) | 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 ph Γ ( 1 4 + 1 2 i t ) assumes its principal value. Because | Z ( t ) | = | ζ ( 1 2 + i t ) | , Z ( t ) vanishes at the zeros of ζ ( 1 2 + i t ) , which can be separated by observing sign changes of Z ( t ) . Because Z ( t ) changes sign infinitely often, ζ ( 1 2 + i t ) has infinitely many zeros with t real. … Sign changes of Z ( t ) are determined by multiplying (25.9.3) by exp ( i ϑ ( t ) ) to obtain the Riemann–Siegel formula: …
14: 19.36 Methods of Computation
§19.36(ii) Quadratic Transformations
Descending Gauss transformations of Π ( ϕ , α 2 , k ) (see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). …If α 2 = k 2 , then the method fails, but the function can be expressed by (19.6.13) in terms of E ( ϕ , k ) , for which Neuman (1969b) uses ascending Landen transformations. … Quadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)). …The function el2 ( x , k c , a , b ) 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 κ , the zonal polynomial Z κ : 𝒮 is defined by the properties
35.4.2 Z κ ( I ) = | κ | !  2 2 | κ | [ m / 2 ] κ 1 j < l ( κ ) ( 2 k j - 2 k l - j + l ) j = 1 ( κ ) ( 2 k j + ( κ ) - j ) !
Therefore Z κ ( T ) is a symmetric polynomial in the eigenvalues of T . … For k = 0 , 1 , 2 , , … For T Ω and ( a ) , ( b ) > 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 { } 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 , 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 1 2 = ( 1 2 , , 1 2 ) . … The number of terms in T N can be greatly reduced by using variables Z = 1 - ( z / A ) with A chosen to make E 1 ( Z ) = 0 . …
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 θ 1 θ θ 2 , | z | Z , where θ = ph z , θ 2 - θ 1 < π , 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 X X Z 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 R S 4 2 ( 3 ) of the ranks 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.