Complex solutions of $z^n = w$, where $n \in \mathbb N^*$ and $w \in \mathbb C.$

Question

I am making this an "Answer your own question" thread, because I see a lot of people asking help with similar exercises :

Let's assume the problem :

Find all the solutions of the equation $z^n = w$, where $n \in \mathbb N^*$ and $w \in \mathbb C.$

The solution of it is listed in the "Answers" area.

Answer 1

There is a standard way which you handle this type of problems.

Solution :

Let $ w=|w|e^{iφ} = |w|\cosφ + i|w|\sin φ, φ=\arg w$

If $ z=|z|e^{iθ} = |z|\cosθ + i|z|\sin θ, θ=\arg z$

Then, $|z|^ne^{inθ}=|w|e^{iφ} \Leftrightarrow |z|^n(|z|\cosθ + i|z|\sin θ)=|w|(\cosφ + i\sin φ)$

Obviously, $|z|^n=|w| \Leftrightarrow |z| = \sqrt[n]{|w|}$

We get :

$|w|e^{inθ} = |w|e^{iφ} \Leftrightarrow |w|(\cos nθ + i\sin nθ) = |w|(\cosφ + i\sin φ) \Rightarrow \{\cos nθ = \cos φ , \sin nθ = \sin φ \} \Rightarrow \{nθ = 2κπ + φ \} \Leftrightarrow \{ θ = \frac{2κπ +φ}{n} \}$

Notice that we reject the solution ${nθ=2κπ - φ}.$

Eventually, the solutions are :

$ |z| = \sqrt[n]{|w|}[\cos (\frac{2κπ +φ}{n}) + i\sin (\frac{2κπ +φ}{n})] = \sqrt[n]{|w|}e^{\frac{2κπ +φ}{n}i} , κ \in \mathbb Z, κ=0,1,...,n-1$ or $κ=1,2,...n $