cubic equations

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2: 1.11 Zeros of Polynomials

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Cubic Equations

… ►For the roots

\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}

g(w)=0

and the roots

\theta_{1},\theta_{2},\theta_{3}

of the resolvent cubic equation … ►

1.11.20

\sqrt{-\theta_{1}}\;\sqrt{-\theta_{2}}\;\sqrt{-\theta_{3}}=-q.

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Symbols:: $q$ and $\theta_{j}$ : cubic roots
Permalink:: http://dlmf.nist.gov/1.11.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §1.11(iii), §1.11(iii), §1.11 and Ch.1

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3: 23.3 Differential Equations

… ►The lattice roots satisfy the cubic equation …

4: 23.22 Methods of Computation

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(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})}},

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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.

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5: 3.8 Nonlinear Equations

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3.8.3

\left|z_{n+1}-\zeta\right|<A{\left|z_{n}-\zeta\right|}^{p}

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Symbols:: $\zeta$ : fixed point
A&S Ref:: 3.9.2
Referenced by:: §3.8(i)
Permalink:: http://dlmf.nist.gov/3.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §3.8(i), §3.8 and Ch.3

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6: 19.29 Reduction of General Elliptic Integrals

… ►These theorems reduce integrals over a real interval

(y,x)

of certain integrands containing the square root of a quartic or cubic polynomial to symmetric integrals over

(0,\infty)

containing the square root of a cubic polynomial (compare §19.16(i)). …Cubic cases of these formulas are obtained by setting one of the factors in (19.29.3) equal to 1. … ►In the cubic case (

h=3

) the basic integrals are … ►(This shows why

I(\mathbf{e}_{\alpha})

is not needed as a basic integral in the cubic case.) … ►In the cubic case, in which

a_{2}=1

b_{2}=0

, (19.29.26) reduces further to …

7: 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). …

8: Bibliography J

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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.

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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

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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.

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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

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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.

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

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

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

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

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

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9: 15.8 Transformations of Variable

… ►In the equations that follow in this subsection all functions take their principal values. … ►

§15.8(v) Cubic Transformations

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Ramanujan’s Cubic Transformation

… ►This is used in a cubic analog of the arithmetic-geometric mean. … ►

10: Bibliography R

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

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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

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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.

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

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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.

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

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

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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

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

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

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