bilinear transformation

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1: 23.15 Definitions

… ►Also $𝒜$ denotes a bilinear transformation on $\tau$, given by ►

23.15.3 $$𝒜\tau =\frac{a\tau +b}{c\tau +d},$$

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Symbols:

$\tau$: complex variable, $𝒜$: bilinear transformation, $a$: integer, $b$: integer, $c$: integer and $d$: integer

Referenced by:

§20.11(i), §23.18

Permalink:

http://dlmf.nist.gov/23.15.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(i), §23.15 and Ch.23

… ►The set of all bilinear transformations of this form is denoted by SL$(2,\mathbb{Z})$ (Serre (1973, p. 77)). … ►

23.15.5 $$f(𝒜\tau )=c_{𝒜}{(c\tau +d)}^{\mathrm{\ell }}f(\tau ),$$ $\mathrm{\Im }\tau >0$,

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Symbols:

$\mathrm{\Im }$: imaginary part, $\tau$: complex variable, $𝒜$: bilinear transformation, $c$: integer, $d$: integer, $f(\tau )$: modular function, $c_{𝒜}$: constant and $\mathrm{\ell }$: level

Permalink:

http://dlmf.nist.gov/23.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(i), §23.15 and Ch.23

…

2: 23.18 Modular Transformations

… ►

23.18.3 $$\lambda \left(𝒜\tau \right)=\lambda \left(\tau \right),$$

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Symbols:

$\lambda \left(\tau \right)$: elliptic modular function, $\tau$: complex variable and $𝒜$: bilinear transformation

Permalink:

http://dlmf.nist.gov/23.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.18, §23.18 and Ch.23

… ►

23.18.4 $$J\left(𝒜\tau \right)=J\left(\tau \right).$$

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Symbols:

$J\left(\tau \right)$: Klein’s complete invariant, $\tau$: complex variable and $𝒜$: bilinear transformation

Referenced by:

§23.18

Permalink:

http://dlmf.nist.gov/23.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §23.18, §23.18 and Ch.23

… ►

23.18.5 $$\eta \left(𝒜\tau \right)=\epsilon (𝒜){\left(-\mathrm{i}(c\tau +d)\right)}^{1/2}\eta \left(\tau \right),$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\mathrm{i}$: imaginary unit, $\tau$: complex variable, $𝒜$: bilinear transformation, $c$: integer, $d$: integer and $\epsilon (𝒜)$: function

Permalink:

http://dlmf.nist.gov/23.18.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.18, §23.18 and Ch.23

… ►

23.18.6 $$\epsilon (𝒜)=\exp \left(\pi \mathrm{i}\left(\frac{a+d}{12c}+s(-d,c)\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $𝒜$: bilinear transformation, $a$: integer, $c$: integer, $d$: integer and $\epsilon (𝒜)$: function

Permalink:

http://dlmf.nist.gov/23.18.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.18, §23.18 and Ch.23

…

3: 32.2 Differential Equations

… ►They are distinct modulo Möbius (bilinear) transformations … ►

32.2.25 $$w(z;\alpha )=ϵW(\zeta )+\frac{1}{{ϵ}^5},$$

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Symbols:

$z$: real, $\alpha$: arbitrary constant and $W(\zeta )$: bilinear transformation

Permalink:

http://dlmf.nist.gov/32.2.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §32.2(vi), §32.2 and Ch.32

… ►

32.2.27 $$\frac{d^{2}W}{{d\zeta }^2}=6W^2+\zeta +{ϵ}^6(2W^3+\zeta W);$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $W(\zeta )$: bilinear transformation

Permalink:

http://dlmf.nist.gov/32.2.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §32.2(vi), §32.2 and Ch.32

… ►

32.2.28 $$w(z;\alpha ,\beta ,\gamma ,\delta )=1+2ϵW(\zeta ;a),$$

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Symbols:

$z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant, $\delta$: arbitrary constant and $W(\zeta )$: bilinear transformation

Permalink:

http://dlmf.nist.gov/32.2.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §32.2(vi), §32.2 and Ch.32

… ►

32.2.30 $$w(z;\alpha ,\beta )=2^{2/3}{ϵ}^{-1}W(\zeta ;a)+{ϵ}^{-3},$$

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Symbols:

$z$: real, $\alpha$: arbitrary constant, $\beta$: arbitrary constant and $W(\zeta )$: bilinear transformation

Permalink:

http://dlmf.nist.gov/32.2.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §32.2(vi), §32.2 and Ch.32

…

4: 1.9 Calculus of a Complex Variable

… ►

Bilinear Transformation

… ►The cross ratio of $z_1,z_2,z_3,z_4\in ℂ\cup \{\mathrm{\infty }\}$ 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

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