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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 ) ,
3: 32.2 Differential Equations
They are distinct modulo Möbius (bilinear) transformations
32.2.25 w ( z ; α ) = ϵ W ( ζ ) + 1 ϵ 5 ,
32.2.27 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 ,
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
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