General Complex Roots

Level 1 - Math I (Physics) topic page in Complex Equations.

Principle

The equation \(z^n=a\) asks for all complex numbers whose \(n\)-th power is a given non-zero complex number. Polar form turns this into matching moduli and arguments.

General root equation

\[z^n=a,\quad a\in\mathbb C,\quad a\ne0\]

A degree \(n\) equation of this form has \(n\) distinct complex roots.

Notation

\(n\) is a positive integer.
\(a=\rho e^{i\chi}\) is the target complex number in polar form.
\(\rho=|a|\) is the modulus of \(a\), so \(\rho\gt0\).
\(\chi\) is an argument of \(a\).
\(z=re^{i\theta}\) is the unknown root in polar form.
\(k\) is an integer that records argument shifts by multiples of \(2\pi\).

Method

Write both sides in polar form:

Both sides polar

\[(re^{i\theta})^n=\rho e^{i\chi}\]

Raise the left-hand side to the \(n\)-th power:

Power expansion

\[r^ne^{in\theta}=\rho e^{i\chi}\]

Match moduli:

Match modulus

\[r^n=\rho\quad\Longrightarrow\quad r=\rho^{1/n}\]

Match arguments, allowing multiples of \(2\pi\):

Match argument

\[n\theta=\chi+2\pi k\]

Therefore:

General argument

\[\theta=\frac{\chi+2\pi k}{n}=\frac{\chi}{n}+\frac{2\pi k}{n}\]

Rules

General roots formula

\[z_k=\rho^{1/n}e^{i(\chi/n+2\pi k/n)},\quad k=0,1,2,\ldots,n-1\]

The roots all have the same modulus \(\rho^{1/n}\).
Their arguments are spaced by \(2\pi/n\).
Increasing \(k\) by \(n\) repeats a root.
The formula assumes \(a\ne0\). If \(a=0\), then \(z=0\) is the only root.

Examples

Solve:

Example equation

\[z^3=1+i\]

First write \(1+i\) in polar form:

Target polar form

\[1+i=\sqrt2 e^{i\pi/4}\]

Here \(n=3\), \(\rho=\sqrt2\), and \(\chi=\pi/4\). The root modulus is:

Example modulus

\[\rho^{1/3}=(\sqrt2)^{1/3}=2^{1/6}\]

The three independent arguments are:

Example arguments

\[\frac{\pi}{12},\quad \frac{\pi}{12}+\frac{2\pi}{3}=\frac{3\pi}{4},\quad \frac{\pi}{12}+\frac{4\pi}{3}=\frac{17\pi}{12}\]

Therefore the three roots are:

Example roots

\[2^{1/6}e^{i\pi/12},\quad 2^{1/6}e^{3\pi i/4},\quad 2^{1/6}e^{17\pi i/12}\]

Checks

Put the target number \(a\) into polar form before taking roots.
Include the \(2\pi k\) term before dividing the argument by \(n\).
List exactly \(n\) independent roots for \(a\ne0\).
Check that all roots lie on the circle of radius \(\rho^{1/n}\).

Roots of Unity

Fundamental Theorem