cubic

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.

…

16: 3.11 Approximation Techniques

… ►A cubic Bézier curve is defined by four points. …The curve is described by

x(t)

and

y(t)

, which are cubic polynomials with

t\in[0,1]

. …

17: 3.4 Differentiation

… ►For additional formulas involving values of

\nabla^{2}u

and

\nabla^{4}u

on square, triangular, and cubic grids, see Collatz (1960, Table VI, pp. 542–546). …

18: Bibliography B

… ►

J. M. Borwein and P. B. Borwein (1991) A cubic counterpart of Jacobi’s identity and the AGM. Trans. Amer. Math. Soc. 323 (2), pp. 691–701.

ⓘ

External Links:: ISSN 0002-9947, FullText, MathReview (Bruce C. Berndt), ZentralBlatt
Referenced by:: §15.8(v).
Encodings:: BibTeX

…

19: Bibliography C

… ►

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

…

20: 19.2 Definitions

… ►Let

s^{2}(t)

be a cubic or quartic polynomial in

t

with simple zeros, and let

r(s,t)

be a rational function of

s

and

t

containing at least one odd power of

s

. …