4 Elementary Functions Applications 4.42 Solution of Triangles 4.44 Other Applications

§4.43 Cubic Equations

ⓘ

Keywords:

applications, cubic equations, solutions as trigonometric and hyperbolic functions, trigonometric functions

Notes:

See Hobson (1928, p. 335).

Referenced by:

§1.11(iii), §3.8(iv), Erratum (V1.0.10) for Section 4.43

Permalink:

http://dlmf.nist.gov/4.43

Errata (effective with 1.0.10):

The first paragraph of this section has been rewritten to correct reported errors. The constants

C=(-\frac{27q^{2}}{4p^{3}})^{1/2}

and

D=-(\frac{27q^{2}}{4p^{3}})^{1/2}

, formerly given in Eq. (4.43.1) for use in Cases (a), (b), and (c) of Eq. (4.43.2), have been eliminated, and these three cases have been corrected. Formerly they read

(a)

$A\sin a$ , $A\sin\left(a+\frac{2}{3}\pi\right)$ , and $A\sin\left(a+\frac{4}{3}\pi\right)$ , with $\sin\left(3a\right)=C$ , when $p<0$ and $C\leq 1$ .
(b)

$A\cosh a$ , $A\cosh\left(a+\frac{2}{3}\pi i\right)$ , and $A\cosh\left(a+\frac{4}{3}\pi i\right)$ , with $\cosh\left(3a\right)=C$ , when $p<0$ and $C>1$ .
(c)

$B\sinh a$ , $B\sinh\left(a+\frac{2}{3}\pi i\right)$ , and $B\sinh\left(a+\frac{4}{3}\pi i\right)$ , with $\sinh\left(3a\right)=D$ , when $p>0$ .

Reported 2014-10-31 by Masataka Urago

See also:

Annotations for Ch.4

Let $p(\neq 0)$ and $q$ be real constants and

4.43.1	$\displaystyle A$	$\displaystyle=\left(-\tfrac{4}{3}p\right)^{1/2},$
4.43.1	$\displaystyle B$	$\displaystyle=\left(\tfrac{4}{3}p\right)^{1/2}.$
ⓘ Symbols: $p$ : real constant and $q$ : real constant Referenced by: (4.43.2), §4.43 Permalink: http://dlmf.nist.gov/4.43.E1 Encodings: TeX, TeX, pMML, pMML, png, png Errata (effective with 1.0.10): The constants $C=(-\frac{27q^{2}}{4p^{3}})^{1/2}$ and $D=-(\frac{27q^{2}}{4p^{3}})^{1/2}$ , formerly given in this equation for use in Cases (a), (b), and (c) of Eq. (4.43.2), have been eliminated, and these three cases have been corrected. Reported 2014-10-31 by Masataka Urago See also: Annotations for §4.43 and Ch.4

The roots of

4.43.2		$z^{3}+pz+q=0$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $a$ : real or complex constant, $z$ : complex variable, $p$ : real constant and $q$ : real constant Referenced by: (4.43.1), §4.43 Permalink: http://dlmf.nist.gov/4.43.E2 Encodings: TeX, pMML, png Errata (effective with 1.0.10): Cases (a), (b), and (c) of of this equation have been corrected. Formerly, with constants $C$ and $D$ given in Eq. (4.43.1), they read (a) $A\sin a$ , $A\sin\left(a+\frac{2}{3}\pi\right)$ , and $A\sin\left(a+\frac{4}{3}\pi\right)$ , with $\sin\left(3a\right)=C$ , when $p<0$ and $C\leq 1$ . (b) $A\cosh a$ , $A\cosh\left(a+\frac{2}{3}\pi i\right)$ , and $A\cosh\left(a+\frac{4}{3}\pi i\right)$ , with $\cosh\left(3a\right)=C$ , when $p<0$ and $C>1$ . (c) $B\sinh a$ , $B\sinh\left(a+\frac{2}{3}\pi i\right)$ , and $B\sinh\left(a+\frac{4}{3}\pi i\right)$ , with $\sinh\left(3a\right)=D$ , when $p>0$ . Reported 2014-10-31 by Masataka Urago See also: Annotations for §4.43 and Ch.4

are:

(a)

$A\sin a$ , $A\sin\left(a+\frac{2}{3}\pi\right)$ , and $A\sin\left(a+\frac{4}{3}\pi\right)$ , with $\sin\left(3a\right)=4q/A^{3}$ , when $4p^{3}+27q^{2}\leq 0$ .
(b)

$A\cosh a$ , $A\cosh\left(a+\frac{2}{3}\pi i\right)$ , and $A\cosh\left(a+\frac{4}{3}\pi i\right)$ , with $\cosh\left(3a\right)=-4q/A^{3}$ , when $p<0$ , $q<0$ , and $4p^{3}+27q^{2}>0$ .
(c)

$B\sinh a$ , $B\sinh\left(a+\frac{2}{3}\pi i\right)$ , and $B\sinh\left(a+\frac{4}{3}\pi i\right)$ , with $\sinh\left(3a\right)=-4q/B^{3}$ , when $p>0$ .

Note that in Case (a) all the roots are real, whereas in Cases (b) and (c) there is one real root and a conjugate pair of complex roots. See also §1.11(iii).

4.42 Solution of Triangles 4.44 Other Applications

NIST

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov