cubic equation

(0.002 seconds)

1—10 of 15 matching pages

3: 23.22 Methods of Computation

… ►

(a)

In the general case, given by $cd\neq 0$ , we compute the roots $\alpha$ , $\beta$ , $\gamma$ , say, of the cubic equation $4t^{3}-ct-d=0$ ; see §1.11(iii). These roots are necessarily distinct and represent $e_{1}$ , $e_{2}$ , $e_{3}$ in some order.

If $c$ and $d$ are real, and the discriminant is positive, that is $c^{3}-27d^{2}>0$ , then $e_{1}$ , $e_{2}$ , $e_{3}$ can be identified via (23.5.1), and $k^{2}$ , ${k^{\prime}}^{2}$ obtained from (23.6.16).

If $c^{3}-27d^{2}<0$ , or $c$ and $d$ are not both real, then we label $\alpha$ , $\beta$ , $\gamma$ so that the triangle with vertices $\alpha$ , $\beta$ , $\gamma$ is positively oriented and $[\alpha,\gamma]$ is its longest side (chosen arbitrarily if there is more than one). In particular, if $\alpha$ , $\beta$ , $\gamma$ are collinear, then we label them so that $\beta$ is on the line segment $(\alpha,\gamma)$ . In consequence, $k^{2}=(\beta-\gamma)/(\alpha-\gamma)$ , ${k^{\prime}}^{2}=(\alpha-\beta)/(\alpha-\gamma)$ satisfy $\Im k^{2}\geq 0\geq\Im{k^{\prime}}^{2}$ (with strict inequality unless $\alpha$ , $\beta$ , $\gamma$ are collinear); also $|k^{2}|$ , $|{k^{\prime}}^{2}|\leq 1$ .

Finally, on taking the principal square roots of $k^{2}$ and ${k^{\prime}}^{2}$ we obtain values for $k$ and $k^{\prime}$ that lie in the 1st and 4th quadrants, respectively, and $2\omega_{1}$ , $2\omega_{3}$ are given by

23.22.1

2\omega_{1}M\left(1,k^{\prime}\right)=-2i\omega_{3}M\left(1,k\right)=\frac{\pi% }{3}\sqrt{\frac{c(2+k^{2}{k^{\prime}}^{2})({k^{\prime}}^{2}-k^{2})}{d(1-k^{2}{% k^{\prime}}^{2})}},

ⓘ

Symbols:: $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{i}$ : imaginary unit, $\omega_{1}$ , $\omega_{3}$ , $\omega_{2}=-\omega_{1}-\omega_{3}$ : lattice generators, $k$ : modulus and $k^{\prime}$ : complementary modulus
Proof sketch:: Combine (23.6.16), (23.6.17), (22.20.6), (23.3.5)–(23.3.7), and use analytical continuation.
Referenced by:: §23.22(ii)
Permalink:: http://dlmf.nist.gov/23.22.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.22(ii), §23.22(ii), §23.22 and Ch.23

where $M$ denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ( $2\omega_{1}$ , $2\omega_{3}$ ), corresponding to the 2 possible choices of the square root.

…

4: 23.3 Differential Equations

… ►The lattice roots satisfy the cubic equation …

5: 3.8 Nonlinear Equations

… ►The equation to be solved is … ►Sometimes the equation takes the form … ►If

p=2

, then the convergence is quadratic; if

p=3

, then the convergence is cubic, and so on. … ►The rule converges locally and is cubically convergent. …

6: 23.21 Physical Applications

… ►The Weierstrass function

\wp

plays a similar role for cubic potentials in canonical form

g_{3}+g_{2}x-4x^{3}

. … ►

§23.21(ii) Nonlinear Evolution Equations

►Airault et al. (1977) applies the function

\wp

to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. For applications to soliton solutions of the Korteweg–de Vries (KdV) equation see McKean and Moll (1999, p. 91), Deconinck and Segur (2000), and Walker (1996, §8.1). …

7: Bibliography J

… ►

D. S. Jones, M. J. Plank, and B. D. Sleeman (2010) Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical and Computational Biology Series, CRC Press, Boca Raton, FL.

ⓘ

Notes:: Second edition of Jones and Sleeman (2003)
External Links:: ISBN 978-1-4200-8357-6, MathReview Entry, ZentralBlatt
Referenced by:: Profile Brian D. Sleeman.
Encodings:: BibTeX

►

D. S. Jones and B. D. Sleeman (2003) Differential equations and mathematical biology. Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL.

ⓘ

Notes:: A second edition was published in 2010.
External Links:: ISBN 1-58488-296-4, MathReview (Jian Hong Wu), ZentralBlatt
Referenced by:: D. S. Jones, M. J. Plank, and B. D. Sleeman (2010).
Encodings:: BibTeX

… ►

N. Joshi and A. V. Kitaev (2001) On Boutroux’s tritronquée solutions of the first Painlevé equation. Stud. Appl. Math. 107 (3), pp. 253–291.

ⓘ

External Links:: ISSN 0022-2526, Document, MathReview, ZentralBlatt
Referenced by:: §32.12(i).
Encodings:: BibTeX

… ►

G. S. Joyce (1973) On the simple cubic lattice Green function. Philos. Trans. Roy. Soc. London Ser. A 273, pp. 583–610.

ⓘ

External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §31.17(ii).
Encodings:: BibTeX

►

G. S. Joyce (1994) On the cubic lattice Green functions. Proc. Roy. Soc. London Ser. A 445, pp. 463–477.

ⓘ

External Links:: ISSN 0962-8444, FullText, MathReview (Peter J. Forrester), ZentralBlatt
Referenced by:: §31.17(ii), §31.7(i).
Encodings:: BibTeX

…

8: Bibliography R

… ►

W. H. Reid (1972) Composite approximations to the solutions of the Orr-Sommerfeld equation. Studies in Appl. Math. 51, pp. 341–368.

ⓘ

External Links:: MathReview (E. A. Boyd), ZentralBlatt
Referenced by:: (9.13.25), (9.13.26), (9.13.28), (9.13.29), (9.13.32), (9.13.33), (9.13.34), §9.13(ii), §9.13(ii).
Encodings:: BibTeX

►

W. H. Reid (1974a) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.

ⓘ

External Links:: MathReview (T. W. Kao), ZentralBlatt
Referenced by:: §2.8(iii).
Encodings:: BibTeX

►

W. H. Reid (1974b) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.

ⓘ

External Links:: MathReview (T. W. Kao), ZentralBlatt
Referenced by:: §2.8(iii).
Encodings:: BibTeX

… ►

W. H. Reid (1997a) Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

ⓘ

External Links:: ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt
Referenced by:: (9.11.16), (9.11.17), §9.11(iii), §9.11(v).
Encodings:: BibTeX

… ►

È. Ya. Riekstynš (1991) Asymptotics and Bounds of the Roots of Equations (Russian). Zinatne, Riga.

ⓘ

External Links:: ISBN 5-7966-0570-4, MathReview (Guillermo López Lagomasino), ZentralBlatt
Referenced by:: §12.11(ii), §6.13.
Encodings:: BibTeX

…

9: Bibliography C

… ►

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

ⓘ

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

… ►

H. H. Chan (1998) On Ramanujan’s cubic transformation formula for

{}_{2}F_{1}(\frac{1}{3},\frac{2}{3};1;z)

. Math. Proc. Cambridge Philos. Soc. 124 (2), pp. 193–204.

ⓘ

External Links:: ISSN 0305-0041, Document, MathReview (Aleksander Grytczuk), ZentralBlatt
Referenced by:: §15.8(v).
Encodings:: BibTeX

… ►

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

ⓘ

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

… ►

P. A. Clarkson (2003b) The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44 (11), pp. 5350–5374.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §32.8(iv).
Encodings:: BibTeX

… ►

E. A. Coddington and N. Levinson (1955) Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York-Toronto-London.

ⓘ

External Links:: MathReview (M. Zlámal)
Referenced by:: §1.18(ix), §1.18(x).
Encodings:: BibTeX

…

10: Bibliography W

… ►

Z. Wang and R. Wong (2003) Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94 (1), pp. 147–194.

ⓘ

External Links:: ISSN 0029-599X, Document, MathReview (Sergey M. Zagorodnyuk), ZentralBlatt
Referenced by:: §2.9(iii).
Encodings:: BibTeX

►

Z. Wang and R. Wong (2005) Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.

ⓘ

External Links:: ISSN 0025-5718, Document, MathReview (B. G. Pachpatte), ZentralBlatt
Referenced by:: §2.9(iii).
Encodings:: BibTeX

… ►

H. Watanabe (1995) Solutions of the fifth Painlevé equation. I. Hokkaido Math. J. 24 (2), pp. 231–267.

ⓘ

External Links:: ISSN 0385-4035, FullText, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.10(v).
Encodings:: BibTeX

… ►

G. N. Watson (1910) The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.

ⓘ

External Links:: ZentralBlatt
Referenced by:: §15.8(v), §15.8(v).
Encodings:: BibTeX

… ►

G. Wolf (1998) On the central connection problem for the double confluent Heun equation. Math. Nachr. 195, pp. 267–276.

ⓘ

External Links:: ISSN 0025-584X, MathReview, ZentralBlatt
Referenced by:: §31.12.
Encodings:: BibTeX

…