# §4.43 Cubic Equations

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

 4.43.1 $\displaystyle A$ $\displaystyle=\left(-\tfrac{4}{3}p\right)^{1/2},$ $\displaystyle B$ $\displaystyle=\left(\tfrac{4}{3}p\right)^{1/2}.$ 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

The roots of

 4.43.2 $z^{3}+pz+q=0$ Symbols: $\mathop{\cosh\/}\nolimits z$: hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$: hyperbolic sine function, $\mathop{\sin\/}\nolimits z$: 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) $A\mathop{\sin\/}\nolimits a$, $A\mathop{\sin\/}\nolimits\!\left(a+\frac{2}{3}\pi\right)$, and $A\mathop{\sin\/}\nolimits\!\left(a+\frac{4}{3}\pi\right)$, with $\mathop{\sin\/}\nolimits\!\left(3a\right)=C$, when $p<0$ and $C\leq 1$. (b) $A\mathop{\cosh\/}\nolimits a$, $A\mathop{\cosh\/}\nolimits\!\left(a+\frac{2}{3}\pi i\right)$, and $A\mathop{\cosh\/}\nolimits\!\left(a+\frac{4}{3}\pi i\right)$, with $\mathop{\cosh\/}\nolimits\!\left(3a\right)=C$, when $p<0$ and $C>1$. (c) $B\mathop{\sinh\/}\nolimits a$, $B\mathop{\sinh\/}\nolimits\!\left(a+\frac{2}{3}\pi i\right)$, and $B\mathop{\sinh\/}\nolimits\!\left(a+\frac{4}{3}\pi i\right)$, with $\mathop{\sinh\/}\nolimits\!\left(3a\right)=D$, when $p>0$. Reported 2014-10-31 by Masataka Urago

are:

1. (a)

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

2. (b)

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

3. (c)

$B\mathop{\sinh\/}\nolimits a$, $B\mathop{\sinh\/}\nolimits\!\left(a+\frac{2}{3}\pi i\right)$, and $B\mathop{\sinh\/}\nolimits\!\left(a+\frac{4}{3}\pi i\right)$, with $\mathop{\sinh\/}\nolimits\!\left(3a\right)=-4q/B^{3}$, 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).