Digital Library of Mathematical Functions

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4 Elementary Functions Applications 4.42 Solution of Triangles 4.44 Other Applications

§4.43 Cubic Equations

Notes:: See Hobson (1928, p. 335).
Keywords:: cubic equations, trigonometric functions
Referenced by:: ¶ ‣ §1.11(iii), §3.8(iv)
Permalink:: http://dlmf.nist.gov/4.43

Let

4.43.1

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

$B=\left(\tfrac{4}{3}p\right)^{{1/2}},$

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

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

Symbols:: : real constant and : 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 are real constants. The roots of

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

Symbols:: : complex variable, : real constant and : real constant
Permalink:: http://dlmf.nist.gov/4.43.E2
Encodings:: TeX, pMML, png

are:

(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 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 and .
(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 .

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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 4.42 Solution of Triangles 4.44 Other Applications