SL(2,Z) bilinear transformation
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11: 10.43 Integrals
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βΊLet be defined as in §10.25(ii).
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§10.43(v) Kontorovich–Lebedev Transform
βΊThe Kontorovich–Lebedev transform of a function is defined as … βΊFor asymptotic expansions of the direct transform (10.43.30) see Wong (1981), and for asymptotic expansions of the inverse transform (10.43.31) see Naylor (1990, 1996). βΊFor collections of the Kontorovich–Lebedev transform, see Erdélyi et al. (1954b, Chapter 12), Prudnikov et al. (1986b, pp. 404–412), and Oberhettinger (1972, Chapter 5). …12: 25.3 Graphics
13: 1.9 Calculus of a Complex Variable
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Bilinear Transformation
… βΊThe cross ratio of is defined by …or its limiting form, and is invariant under bilinear transformations. βΊOther names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation. …14: 19.36 Methods of Computation
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§19.36(ii) Quadratic Transformations
… βΊDescending Gauss transformations of (see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). …If , then the method fails, but the function can be expressed by (19.6.13) in terms of , for which Neuman (1969b) uses ascending Landen transformations. … βΊQuadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)). …The function is computed by descending Landen transformations if is real, or by descending Gauss transformations if is complex (Bulirsch (1965b)). …15: 25.10 Zeros
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βΊCalculations relating to the zeros on the critical line make use of the real-valued function
…is chosen to make real, and assumes its principal value.
Because , vanishes at the zeros of , which can be separated by observing sign changes of .
Because changes sign infinitely often, has infinitely many zeros with real.
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βΊSign changes of are determined by multiplying (25.9.3) by to obtain the Riemann–Siegel formula:
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16: 35.4 Partitions and Zonal Polynomials
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βΊFor any partition , the zonal polynomial
is defined by the properties
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βΊTherefore is a symmetric polynomial in the eigenvalues of .
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βΊFor ,
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βΊFor and ,
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35.4.8
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17: 10.44 Sums
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10.44.1
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βΊIf and the upper signs are taken, then the restriction on is unnecessary.
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10.44.3
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βΊThe restriction is unnecessary when and is an integer.
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10.44.6
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18: 19.19 Taylor and Related Series
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βΊFor define the homogeneous hypergeometric polynomial
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βΊIf , then (19.19.3) is a Gauss hypergeometric series (see (19.25.43) and (15.2.1)).
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βΊand define the -tuple .
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βΊThe number of terms in can be greatly reduced by using variables with chosen to make .
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19.19.7
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19: 18.38 Mathematical Applications
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βΊIt has elegant structures, including -soliton solutions, Lax pairs, and Bäcklund transformations.
While the Toda equation is an important model of nonlinear systems, the special functions of mathematical physics are usually regarded as solutions to linear equations.
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.
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Radon Transform
… βΊDefine a further operator by …20: Bibliography
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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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Lectures on Integral Transforms.
Translations of Mathematical Monographs, Vol. 70, American Mathematical Society, Providence, RI.
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Transformations
of the ranks and algebraic solutions of the sixth Painlevé equation.
Comm. Math. Phys. 228 (1), pp. 151–176.
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Summations and transformations for basic Appell series.
J. London Math. Soc. (2) 4, pp. 618–622.
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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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