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1: 1.14 Integral Transforms
§1.14 Integral Transforms
§1.14(i) Fourier Transform
§1.14(iii) Laplace Transform
Fourier Transform
Laplace Transform
2: 23.15 Definitions
Also 𝒜 denotes a bilinear transformation on τ , given by
23.15.3 𝒜 τ = a τ + b c τ + d ,
The set of all bilinear transformations of this form is denoted by SL ( 2 , ) (Serre (1973, p. 77)). …
23.15.5 f ( 𝒜 τ ) = c 𝒜 ( c τ + d ) f ( τ ) , τ > 0 ,
3: 23.18 Modular Transformations
§23.18 Modular Transformations
23.18.3 λ ( 𝒜 τ ) = λ ( τ ) ,
23.18.4 J ( 𝒜 τ ) = J ( τ ) .
23.18.5 η ( 𝒜 τ ) = ε ( 𝒜 ) ( i ( c τ + d ) ) 1 / 2 η ( τ ) ,
23.18.6 ε ( 𝒜 ) = exp ( π i ( a + d 12 c + s ( d , c ) ) ) ,
4: 32.2 Differential Equations
They are distinct modulo Möbius (bilinear) transformations
32.2.25 w ( z ; α ) = ϵ W ( ζ ) + 1 ϵ 5 ,
32.2.27 d 2 W d ζ 2 = 6 W 2 + ζ + ϵ 6 ( 2 W 3 + ζ W ) ;
32.2.28 w ( z ; α , β , γ , δ ) = 1 + 2 ϵ W ( ζ ; a ) ,
32.2.30 w ( z ; α , β ) = 2 2 / 3 ϵ 1 W ( ζ ; a ) + ϵ 3 ,
5: Bibliography
  • M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
  • S. V. Aksenov, M. A. Savageau, U. D. Jentschura, J. Becher, G. Soff, and P. J. Mohr (2003) Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics. Comput. Phys. Comm. 150 (1), pp. 1–20.
  • D. E. Amos (1989) Repeated integrals and derivatives of K Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.
  • 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.
  • 6: 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. …
    7: 20 Theta Functions
    Chapter 20 Theta Functions
    8: 18.38 Mathematical Applications
    It has elegant structures, including N -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. …
    Radon Transform
    9: 20.10 Integrals
    §20.10(i) Mellin Transforms with respect to the Lattice Parameter
    20.10.1 0 x s 1 θ 2 ( 0 | i x 2 ) d x = 2 s ( 1 2 s ) π s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
    20.10.2 0 x s 1 ( θ 3 ( 0 | i x 2 ) 1 ) d x = π s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
    20.10.3 0 x s 1 ( 1 θ 4 ( 0 | i x 2 ) ) d x = ( 1 2 1 s ) π s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 0 .
    §20.10(ii) Laplace Transforms with respect to the Lattice Parameter
    10: Bibliography N
  • D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
  • D. Naylor (1990) On an asymptotic expansion of the Kontorovich-Lebedev transform. Applicable Anal. 39 (4), pp. 249–263.
  • D. Naylor (1996) On an asymptotic expansion of the Kontorovich-Lebedev transform. Methods Appl. Anal. 3 (1), pp. 98–108.
  • W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.
  • E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.