Complex Numbers in Solving Cubic Equations

Complex numbers arise naturally from the solving of cubic equations.

Consider, for example, the cubic equation x3+px+q=0x^3+ px + q = 0 (incidentally, all cubic polynomial equations can be rewritten in this form). Solving for xx in terms of pp and qq is possible. The solution is given by:

x=A3+B3A=q2+q24+p327B=q2q24+p327\begin{aligned} x &= \sqrt[3]{A} + \sqrt[3]{B} \\ A &= -\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}} \\ B &= -\frac{q}{2} - \sqrt{\frac{q^2}{4} + \frac{p^3}{27}} \end{aligned}

(Source: Lectures on Elementary Mathematics by Lagrange. Highly recommended.)

This is the cubic analogue of the famous and familiar "quadratic formula" for calculating roots of quadratic polynomials.

The above equation, however, raises two concerns:

  1. All cubic polynomials have a real root (provided the coefficients are real, at least). However, it is clear that if AA and BB will have imaginary components if q24+p327\frac{q^2}{4} + \frac{p^3}{27}.
  2. Cubic polynomials may have up to three roots. But this multiplicity isn't obviously.

We first recall Euler's formula:

eiθ=cos(θ)+sin(θ)i.e^{i \theta} = \cos (\theta) + \sin(\theta) i.

So we can write AA and BB as

A=reiθB=reiθ.\begin{aligned} A &= r e^{i \theta} \\ B &= r e^{-i \theta}. \end{aligned}

for some unique rr and unique θ\theta (unique modulo 2π2\pi).

In the complex field, roots are considered to be multi-valued. Unlike with real roots, we cannot pick the "positive" complex root of xn=Cx^n = C as the value of Cn\sqrt[n]{C}.

A3=r3eθ+2kπ3,\sqrt[3]{A}=\sqrt[3]{r} \cdot e^{\frac{\theta + 2k\pi}{3}},

where k=0,1,2k=0,1,2 give unique roots.

Created 2020-04-28. Last updated 2020-06-22. View source.