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bilinear transformation

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1: 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 ,
2: 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 ) ) ) ,
3: 32.2 Differential Equations
They are distinct modulo Möbius (bilinear) transformations
32.2.25 w ( z ; α ) = ϵ W ( ζ ) + 1 ϵ 5 ,
32.2.28 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 ,
32.2.32 w ( z ; α , β , γ , δ ) = 1 + ϵ ζ W ( ζ ; a , b , c , d ) ,
4: 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. …
5: 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