Question

Let
w=ρ(cosφ+isinφ)
,
z=r(cosθ+isinθ)
and if n is a positive integer, the nth roots of a complex number are by definition the value of w which satisfies the equation

Answer 1

Answer on Question #58325 – Math – Complex Analysis

Question

Let

w=ρ(cosϕ+isinϕ),

z=r(cosθ+isinθ)

and if n is a positive integer, the nth roots of a complex number are by definition the value of w which satisfies the equation

Solution

The nth roots of a complex number $z = r(cos\theta + isin\theta)$ are defined by

$$w = \sqrt[n]{z},$$

where $w = \rho(cos\phi + isin\phi), \rho = \sqrt[n]{r}, \phi = \frac{\theta + 2\pi k}{n}, k = 0, 1, \ldots, n - 1.$

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