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I know how to find the roots to the equation $z^n=w$, for $n \in \mathbb{R\setminus\{ 0\}}$ (by writing $w$ as $re^{i(\theta+2k\pi)}$), and taking the nth root of both sides, which I'm perfectly happy with, since $r^{1/n}$ is easy to find (with a calculator), as is $\frac{\theta+2k\pi}{n}$.

But what happens when we want to solve the equation $$z^t=w \tag{*}$$ for $t \in \mathbb{C}$?

e.g. how would one go about finding all roots of $$z^{2+3i}=1+i?$$

Obviously, it's not simply a case of writing $1+i$ in polar form and then taking the $(2+3i)$th root of $1+i$, since we don't know what $$\left| 1+i\right|^{1/({2+3i})}=\sqrt{2}^{1/(2+3i)}$$ is (or, even, that it exists).

Any ideas?

Short version: how do I find $t$th roots of a complex number for $t \in \mathbb{C}$?

Also, is there any way that we can determine how many roots this $(*)$ will have?

beep-boop
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    Short summary: Say you want $w^z$. This is equal to $\exp{z\cdot \ln w}$, so the only interesting part is how to find $\ln w$; that is, how to find $x$ with $e^{x} = w$. The infinite family of $z^w$ is then completely determined by the infinite family of such $x$, all of which have the form $x = y + 2ik\pi$ for some fixed $y$ and arbitrary integer $k$. – MJD Jun 19 '14 at 19:49
  • (I have elaborated on my comment over at the other post.) – MJD Jun 19 '14 at 20:01

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