$i^{th}$ root of a complex number

Question

So let's say you have $6+2i$ and you want to find $\sqrt[i]{6+2i}$. However what I would like is a more general formula for $$\sqrt[i]{a+bi}$$. I got this far $\sqrt[i]{a+bi}= e^{-iln(a+bi)}$ How do I continue?

Answer 1

The $i$th root of a complex number $z$ would be expressed as $z^{1/i}$, or $z^{-i}$. This, in turn, can be expressed as $$z^{-i}=\exp(-i\log z),$$ where $\log$ is interpreted as a suitable inverse of the exponential function. Since we can choose different branches of $\log$, we end up being able to choose different branches of the $i$th root function.

Answer 2

In complex analysis one can define any power as $a^b=e^{b\log(a)}$ whenever $\log(a)$ is defined. The exact value of $\log(a)$ depends on how you choose your branch cut. For positive real $a$ there is a canonical choice, but otherwise it depends on what you are trying to achieve.

You will need the logarithm of $6+2i$ and you can then calculate $$ \sqrt[i]{6+2i} = (6+2i)^{1/i} = (6+2i)^{-i} = e^{-i\log(6+2i)}. $$ The multiple possible values of the logarithm correspond to having several roots.