# §4.43 Cubic Equations

Let

 4.43.1 $\displaystyle A$ $\displaystyle=\left(-\tfrac{4}{3}p\right)^{1/2},$ $\displaystyle B$ $\displaystyle=\left(\tfrac{4}{3}p\right)^{1/2},$ $\displaystyle C$ $\displaystyle=\left(-\frac{27q^{2}}{4p^{3}}\right)^{1/2},$ $\displaystyle D$ $\displaystyle=-\left(\frac{27q^{2}}{4p^{3}}\right)^{1/2},$ Symbols: $p$: real constant and $q$: real constant Permalink: http://dlmf.nist.gov/4.43.E1 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

where $p(\neq 0)$ and $q$ are real constants. The roots of

 4.43.2 $z^{3}+pz+q=0$

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 $\mathop{\sin\/}\nolimits\!\left(3a\right)=C$, when $p<0$ and $C\leq 1$.

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 $\mathop{\cosh\/}\nolimits\!\left(3a\right)=C$, when $p<0$ and $C>1$.

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 $\mathop{\sinh\/}\nolimits\!\left(3a\right)=D$, 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).