linear transformation

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1: 15.19 Methods of Computation

… ►For

z\in\mathbb{R}

it is always possible to apply one of the linear transformations in §15.8(i) in such a way that the hypergeometric function is expressed in terms of hypergeometric functions with an argument in the interval

[0,\frac{1}{2}]

. ►For

z\in\mathbb{C}

it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when

z={\mathrm{e}}^{\pm\pi\mathrm{i}/3}

. This is because the linear transformations map the pair

\{{\mathrm{e}}^{\pi\mathrm{i}/3},{\mathrm{e}}^{-\pi\mathrm{i}/3}\}

onto itself. … ►When

\Re z>\frac{1}{2}

it is better to begin with one of the linear transformations (15.8.4), (15.8.7), or (15.8.8). … …

2: 1.9 Calculus of a Complex Variable

… ►The linear transformation

f(z)=az+b

a\not=0

, has

f^{\prime}(z)=a

and

w=f(z)

maps

\mathbb{C}

conformally onto

\mathbb{C}

. ►

Bilinear Transformation

… ►Other names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation. …

3: 15.8 Transformations of Variable

… ►

§15.8(i) Linear Transformations

… ►

§15.8(ii) Linear Transformations: Limiting Cases

… ►

15.8.12

\mathbf{F}\left(a,b;a+b-m;z\right)=(1-z)^{-m}\mathbf{F}\left(\tilde{a},\tilde{% b};\tilde{a}+\tilde{b}+m;z\right),

\tilde{a}=a-m,\tilde{b}=b-m

ⓘ

Symbols:: $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $m$ : integer
Permalink:: http://dlmf.nist.gov/15.8.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §15.8(ii), §15.8 and Ch.15

… ►A quadratic transformation relates two hypergeometric functions, with the variable in one a quadratic function of the variable in the other, possibly combined with a fractional linear transformation. … ►The transformation formulas between two hypergeometric functions in Group 2, or two hypergeometric functions in Group 3, are the linear transformations (15.8.1). …

4: 18.7 Interrelations and Limit Relations

… ►

§18.7(i) Linear Transformations

…

5: Bibliography S

… ►

D. Shanks (1955) Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.

ⓘ

External Links:: MathReview (J. Kuntzmann), ZentralBlatt
Referenced by:: §17.18.
Encodings:: BibTeX

… ►

M. H. Stone (1990) Linear transformations in Hilbert space. American Mathematical Society Colloquium Publications, Vol. 15, American Mathematical Society, Providence, RI.

ⓘ

Notes:: Reprint of the 1932 original
External Links:: ISBN 0-8218-1015-4, Document, Link, MathReview (P. R. Halmos)
Referenced by:: §1.18(x).
Encodings:: BibTeX

…

6: 15.12 Asymptotic Approximations

… ►By combination of the foregoing results of this subsection with the linear transformations of §15.8(i) and the connection formulas of §15.10(ii), similar asymptotic approximations for

F\left(a+e_{1}\lambda,b+e_{2}\lambda;c+e_{3}\lambda;z\right)

can be obtained with

e_{j}=\pm 1

0

j=1,2,3

. …

7: 19.14 Reduction of General Elliptic Integrals

… ►The last reference gives a clear summary of the various steps involving linear fractional transformations, partial-fraction decomposition, and recurrence relations. …

8: Mark J. Ablowitz

… ►for appropriate data they can be linearized by the Inverse Scattering Transform (IST) and they possess solitons as special solutions. …

9: Bibliography E

… ►

W. N. Everitt (1982) On the transformation theory of ordinary second-order linear symmetric differential expressions. Czechoslovak Math. J. 32(107) (2), pp. 275–306.

ⓘ

Notes:: With a loose Russian summary
External Links:: ISSN 0011-4642, MathReview (D. Hinton)
Referenced by:: §1.13(viii).
Encodings:: BibTeX

…

10: Bruce R. Miller

… ►There, he carried out research in non-linear dynamics and celestial mechanics, developing a specialized computer algebra system for high-order Lie transformations. …