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1: 4.43 Cubic Equations

§4.43 Cubic Equations

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2: 16.6 Transformations of Variable

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Cubic

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3: 1.11 Zeros of Polynomials

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Cubic Equations

… ►For the roots ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4$ of $g(w)=0$ 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

… ►Resolvent cubic is $z^3+12z^2+20z+9=0$ with roots ${\theta }_1=-1$, ${\theta }_2=-\frac{1}{2}(11+\sqrt{85})$, ${\theta }_3=-\frac{1}{2}(11-\sqrt{85})$, and $\sqrt{-{\theta }_1}=1$, $\sqrt{-{\theta }_2}=\frac{1}{2}(\sqrt{17}+\sqrt{5})$, $\sqrt{-{\theta }_3}=\frac{1}{2}(\sqrt{17}-\sqrt{5})$. …

4: 3.8 Nonlinear Equations

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3.8.3

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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

… ►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. …

5: Bibliography J

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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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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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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 $(0,\mathrm{\infty })$ 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 $I({𝐞}_{\alpha })$ 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: 15.8 Transformations of Variable

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§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. … ►

8: 23.21 Physical Applications

… ►The Weierstrass function $\mathrm{\wp }$ plays a similar role for cubic potentials in canonical form $g_3+g_{2}x-4x^3$. …

9: 10.74 Methods of Computation

… ►Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent. …

10: 19.14 Reduction of General Elliptic Integrals

… ►The choice among 21 transformations for final reduction to Legendre’s normal form depends on inequalities involving the limits of integration and the zeros of the cubic or quartic polynomial. …