# How do you determine the principal root of a unit complex number?

Let's suppose t be in the inverval $$(-\pi, \pi]$$ and that $$n$$ is a natural number. What is $$(\cos t + i\sin t )^{\frac 1n}$$? Using Euler's formula would give us the following:

$$(\cos t + i\sin t )^{\frac 1n}=$$

$$(e^{it})^{\frac 1n}=$$

$$e^{it\times \frac 1n} =$$

$$e^{\frac{it}{n}} =$$

$$\cos \frac 1n + i\sin \frac 1n$$.

However, this would be problematic if we're taking on odd root of $$-1$$. When we're just dealing with the real numbers, any odd root of $$-1$$ is $$-1$$. However, if $$n$$ is odd, then based on formula above, since the argument of $$-1$$ is $$\pi$$, $$(-1)^{\frac 1n} = \cos \frac{\pi}{n}+ i\sin \frac{\pi}{n}$$. So $$(-1)^{\frac 13}$$ would be equal to $$\frac 12 + i \frac{\sqrt 3}{2}$$, even though it would just be $$-1$$ if we were only dealing with the real numbers. It doesn't make sense that the same operation would yield a different result just because we've extended the number-system we're working with. So, I was wondering if this in fact, is the correct formula for determining the principal root of a unti complex number.

$$(\cos t + i\sin t )^{\frac 1n}= \cos \frac 1n + i\sin \frac 1n$$ if $$-\pi or $$n$$ is even

$$=-1$$ if $$t=\pi$$ and $$n$$ is odd

If this isn't the correct formula, then what is the correct formula?

This is the reason why one should never talk about "the" root of a number when dealing with complex numbers -1 has 3 cube roots, $$\frac{1}{2}+i\frac{\sqrt{3}}{2},-1$$ and $$\frac{1}{2}-i\frac{\sqrt{3}}{2}$$. Your method only gives the root with the smallest argument which is $$\frac{1}{2}+i\frac{\sqrt{3}}{2}$$.
I could be wrong, but I don't think there is a universally accepted definition of "principal root" in the context of complex analysis. Since we are working over an algebraically closed field, $$\mathbb{C},$$ the equation $$z^n-z_0=0$$ for $$z_0$$ having unit radius (or any modulus for that matter) has $$n$$ solutions.
The selection of one of these $$n$$-th roots as being special somehow is more arbitrary that (e.g.) the selection of a square root of a positive number in $$\mathbb{R}.$$ since there aren't really notions of positive and negative in the same way.
One could do something by forcing all the angles of the roots between $$[0,2\pi)$$ and then select the root with the smallest angle