quadratic equations

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11: Bibliography C

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R. Campbell (1955) Théorie Générale de L’Équation de Mathieu et de quelques autres Équations différentielles de la mécanique. Masson et Cie, Paris (French).

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §28.1, Notations C, Notations I, Notations I, Notations J, Notations J, Notations S.
Encodings:: BibTeX

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B. C. Carlson and J. FitzSimons (2000) Reduction theorems for elliptic integrands with the square root of two quadratic factors. J. Comput. Appl. Math. 118 (1-2), pp. 71–85.

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Notes:: Code written by the second author to compute symmetric elliptic integrals numerically in the Derive computer algebra system is available.
External Links:: ISSN 0377-0427, Document, Code, MathReview, ZentralBlatt
Referenced by:: §19.29(ii), §19.36(i), §19.36(i), §19.39(iv).
Encodings:: BibTeX

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B. C. Carlson (1976) Quadratic transformations of Appell functions. SIAM J. Math. Anal. 7 (2), pp. 291–304.

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External Links:: Document, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §16.16(ii), §19.22(i).
Encodings:: BibTeX

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T. W. Chaundy (1969) Elementary Differential Equations. Clarendon Press, Oxford.

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External Links:: ISBN 0-19-853139-7, MathReview, ZentralBlatt
Referenced by:: §16.12.
Encodings:: BibTeX

… ►

J. Choi and A. K. Rathie (2013) An extension of a Kummer’s quadratic transformation formula with an application. Proc. Jangjeon Math. Soc. 16 (2), pp. 229–235.

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External Links:: ISSN 1598-7264, MathReview (Petr Blaschke), ZentralBlatt
Referenced by:: §16.6, §16.6.
Encodings:: BibTeX

…

12: Bibliography G

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W. Gautschi (1997b) The Computation of Special Functions by Linear Difference Equations. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 213–243.

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External Links:: ISBN 90-5699-521-9, MathReview, ZentralBlatt
Referenced by:: §3.6(iii), §3.6(vii).
Encodings:: BibTeX

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Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:: ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

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J. J. Gray (2000) Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

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External Links:: ISBN 0-8176-3837-7, MathReview, ZentralBlatt
Referenced by:: §15.17(v).
Encodings:: BibTeX

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V. I. Gromak and N. A. Lukaševič (1982) Special classes of solutions of Painlevé equations. Differ. Uravn. 18 (3), pp. 419–429 (Russian).

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Notes:: In Russian. English translation: Differential Equations 18(3), pp. 317–326.
External Links:: ISSN 0374-0641, MathReview, ZentralBlatt
Referenced by:: §32.10(iv), §32.10(v), §32.8(iii), §32.8(v), §32.9(ii).
Encodings:: BibTeX

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V. I. Gromak (1975) Theory of Painlevé’s equations. Differ. Uravn. 11 (11), pp. 373–376 (Russian).

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Notes:: In Russian. English translation: Differential Equations 11(11), pp. 285–287
External Links:: MathReview (Z. Nehari), ZentralBlatt
Referenced by:: §32.7(iii), §32.7(vi).
Encodings:: BibTeX

…

13: 10.74 Methods of Computation

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§10.74(ii) Differential Equations

►A comprehensive and powerful approach is to integrate the differential equations (10.2.1) and (10.25.1) by direct numerical methods. As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation. … ►For further information, including parallel methods for solving the differential equations, see Lozier and Olver (1993). … ►Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent. …

14: 32.7 Bäcklund Transformations

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§32.7(ii) Second Painlevé Equation

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§32.7(iii) Third Painlevé Equation

… ►

\mbox{P}_{\mbox{\scriptsize VI}}

also has quadratic and quartic transformations. …The quadratic transformation … …

15: 19.22 Quadratic Transformations

§19.22 Quadratic Transformations

… ►As

n\to\infty

p_{n}

and

\varepsilon_{n}

converge quadratically to

M\left(a_{0},g_{0}\right)

and 0, respectively, and

Q_{n}

converges to 0 faster than quadratically. … ►The equations inverse to (19.22.5) and (19.22.16) are given by …and the corresponding equations with

z

z_{+}

, and

z_{-}

replaced by

p

p_{+}

, and

p_{-}

, respectively. …

16: 16.16 Transformations of Variables

… ►

16.16.5_5

{F_{4}}\left(\alpha,\beta;\gamma,\beta;x(1-y),y(1-x)\right)=(1-x)^{-\alpha}(1-% y)^{-\alpha}{F_{1}}\left(\alpha;\gamma-\beta,\alpha-\gamma+1;\gamma;\frac{x}{x% -1},\frac{xy}{(1-x)(1-y)}\right),

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Symbols:: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function and ${F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{\gamma},\NVar{\gamma^{\prime}};% \NVar{x},\NVar{y}\right)$ : fourth Appell function
Referenced by:: §16.16(i), Erratum (V1.1.12) for Additions
Permalink:: http://dlmf.nist.gov/16.16.E5_5
Encodings:: pMML, png, TeX
Addition (effective with 1.1.12):: This equation was added.
See also:: Annotations for §16.16(i), §16.16 and Ch.16

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16.16.10

{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)=\frac{\Gamma\left(% \gamma^{\prime}\right)\Gamma\left(\beta-\alpha\right)}{\Gamma\left(\gamma^{% \prime}-\alpha\right)\Gamma\left(\beta\right)}(-y)^{-\alpha}{F_{4}}\left(% \alpha,\alpha-\gamma^{\prime}+1;\gamma,\alpha-\beta+1;\frac{x}{y},\frac{1}{y}% \right)+\frac{\Gamma\left(\gamma^{\prime}\right)\Gamma\left(\alpha-\beta\right% )}{\Gamma\left(\gamma^{\prime}-\beta\right)\Gamma\left(\alpha\right)}(-y)^{-% \beta}{F_{4}}\left(\beta,\beta-\gamma^{\prime}+1;\gamma,\beta-\alpha+1;\frac{x% }{y},\frac{1}{y}\right).

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Symbols:: ${F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{\gamma},\NVar{\gamma^{\prime}};% \NVar{x},\NVar{y}\right)$ : fourth Appell function and $\Gamma\left(\NVar{z}\right)$ : gamma function
Permalink:: http://dlmf.nist.gov/16.16.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §16.16(ii), §16.16 and Ch.16

►For quadratic transformations of Appell functions see Carlson (1976). …

17: 18.2 General Orthogonal Polynomials

… ►It is assumed throughout this chapter that for each polynomial

p_{n}(x)

that is orthogonal on an open interval

(a,b)

the variable

x

is confined to the closure of

(a,b)

unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.) … ►

18.2.11_2

a_{n-1}c_{n}>0,

n\geq 1

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Symbols:: $n$ : nonnegative integer, $a_{n}$ : real constant and $c_{n}$ : real constant
Referenced by:: §18.2(viii)
Permalink:: http://dlmf.nist.gov/18.2.E11_2
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

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18.2.11_6

\beta_{n}>0,

n\geq 1

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Symbols:: $n$ : nonnegative integer
Referenced by:: §18.2(viii)
Permalink:: http://dlmf.nist.gov/18.2.E11_6
Encodings:: pMML, png, TeX
Addition (effective with 1.2.0):: This equation was added.
See also:: Annotations for §18.2(iv), §18.2(iv), §18.2 and Ch.18

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§18.2(vii) Quadratic Transformations

… ►Equations (18.14.3_5) and (18.14.8), both for

\alpha=0

, can be seen as special cases of this result for Jacobi and Laguerre polynomials, respectively.

18: 31.7 Relations to Other Functions

… ►They are analogous to quadratic and cubic hypergeometric transformations (§§15.8(iii)–15.8(v)). … ►equation (31.2.1) becomes Lamé’s equation with independent variable

\zeta

; compare (29.2.1) and (31.2.8). The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. …

19: Bibliography

… ►

A. S. Abdullaev (1985) Asymptotics of solutions of the generalized sine-Gordon equation, the third Painlevé equation and the d’Alembert equation. Dokl. Akad. Nauk SSSR 280 (2), pp. 265–268 (Russian).

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Notes:: English translation: Soviet Math. Dokl. 31(1985), no. 1, pp. 45–47
External Links:: ISSN 0002-3264, MathReview (M. S. Agranovich), ZentralBlatt
Referenced by:: §32.11(iv).
Encodings:: BibTeX

… ►

V. È. Adler (1994) Nonlinear chains and Painlevé equations. Phys. D 73 (4), pp. 335–351.

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External Links:: ISSN 0167-2789, Document, MathReview (F. Pempinelli), ZentralBlatt
Referenced by:: §32.2(v).
Encodings:: BibTeX

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F. M. Arscott (1967) The Whittaker-Hill equation and the wave equation in paraboloidal co-ordinates. Proc. Roy. Soc. Edinburgh Sect. A 67, pp. 265–276.

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External Links:: MathReview (H. Hochstadt), ZentralBlatt
Referenced by:: §28.31(i), §28.31(ii), §28.31(ii), §28.32(ii).
Encodings:: BibTeX

… ►

U. M. Ascher and L. R. Petzold (1998) Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:: ISBN 0-89871-412-5, MathReview (Michael Hanke), ZentralBlatt
Referenced by:: §3.7(i).
Encodings:: BibTeX

… ►

M. J. Atia, A. Martínez-Finkelshtein, P. Martínez-González, and F. Thabet (2014) Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters. J. Math. Anal. Appl. 416 (1), pp. 52–80.

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External Links:: ISSN 0022-247X, Document, Link, MathReview (Vivek Sahai), ZentralBlatt
Referenced by:: §18.15(vi), §18.15(vi).
Encodings:: BibTeX

…

20: 19.29 Reduction of General Elliptic Integrals

… ►If both square roots in (19.29.22) are 0, then the indeterminacy in the two preceding equations can be removed by using (19.27.8) to evaluate the integral as

R_{G}\left(a_{1}b_{2},a_{2}b_{1},0\right)

multiplied either by

-2/(b_{1}b_{2})

or by

-2/(a_{1}a_{2})

in the cases of (19.29.20) or (19.29.21), respectively. …