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4 Elementary FunctionsApplications

§4.43 Cubic Equations

Let p(0) and q be real constants and

4.43.1 A =(-43p)1/2,
B =(43p)1/2.

The roots of

4.43.2 z3+pz+q=0

are:

  1. (a)

    Asina, Asin(a+23π), and Asin(a+43π), with sin(3a)=4q/A3, when 4p3+27q20.

  2. (b)

    Acosha, Acosh(a+23πi), and Acosh(a+43πi), with cosh(3a)=-4q/A3, when p<0, q<0, and 4p3+27q2>0.

  3. (c)

    Bsinha, Bsinh(a+23πi), and Bsinh(a+43πi), with sinh(3a)=-4q/B3, 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).