# 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(ii) LinearTransformations: Limiting Cases
15.8.12 $\mathbf{F}\left(a,b;a+b-m;z\right)=(1-z)^{-m}\mathbf{F}\left(\tilde{a},\tilde{% b};\tilde{a}+\tilde{b}+m;z\right),$ $\tilde{a}=a-m,\tilde{b}=b-m$.
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). …
##### 5: Bibliography S
• D. Shanks (1955) Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, pp. 1–42.
• M. H. Stone (1990) Linear transformations in Hilbert space. American Mathematical Society Colloquium Publications, Vol. 15, American Mathematical Society, Providence, RI.
• ##### 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 $e_{j}=\pm 1$ or $0$, $j=1,2,3$. …
##### 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.
• ##### 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. …