# bilinear transformation

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## 5 matching pages

##### 1: 23.15 Definitions
βΊAlso $\mathcal{A}$ denotes a bilinear transformation on $\tau$, given by βΊ βΊThe set of all bilinear transformations of this form is denoted by SL$(2,\mathbb{Z})$ (Serre (1973, p. 77)). … βΊ
23.15.5 $f(\mathcal{A}\tau)=c_{\mathcal{A}}(c\tau+d)^{\ell}f(\tau),$ $\Im\tau>0$,
##### 2: 23.18 Modular Transformations
βΊ βΊ βΊ
23.18.5 $\eta\left(\mathcal{A}\tau\right)=\varepsilon(\mathcal{A})\left(-i(c\tau+d)% \right)^{1/2}\eta\left(\tau\right),$
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23.18.6 $\varepsilon(\mathcal{A})=\exp\left(\pi i\left(\frac{a+d}{12c}+s(-d,c)\right)% \right),$
##### 3: 32.2 Differential Equations
βΊThey are distinct modulo Möbius (bilinear) transformationsβΊ
32.2.25 $w(z;\alpha)=\epsilon W(\zeta)+\frac{1}{\epsilon^{5}},$
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32.2.28 $w(z;\alpha,\beta,\gamma,\delta)=1+2\epsilon W(\zeta;a),$
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##### 4: 1.9 Calculus of a Complex Variable
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###### BilinearTransformation
βΊThe cross ratio of $z_{1},z_{2},z_{3},z_{4}\in\mathbb{C}\cup\{\infty\}$ is defined by …or its limiting form, and is invariant under bilinear transformations. βΊOther names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation. …
##### 5: 18.38 Mathematical Applications
βΊIt 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. … βΊ