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Could anyone give me some explanation how can I compute sum of multibranched functions? For example, if I have $z=re^{i\varphi}$ then $\sqrt{z}=\sqrt{r}(\cos(\varphi/2+k\pi)+i\sin (\varphi/2+k\pi))$ for some integer $k$. Therefore, is $\sqrt{z}+\sqrt{z}=2\sqrt{r}(\cos(\varphi/2+k\pi)+i\sin (\varphi/2+k\pi))$ for some integer $k$ or $\sqrt{z}+\sqrt{z}=\sqrt{r}(\cos(\varphi/2+k_1\pi)+i\sin (\varphi/2+k_1\pi))+\sqrt{r}(\cos(\varphi/2+k_2\pi)+i\sin (\varphi/2+k_2\pi))$ for some integers $k_1,k_2$?

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    @student: I read your last comment because I happened to come back to this page. If you want people to be notified of your responses, you need to put an '@' in front of their name.2011-10-22

2 Answers 2

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The notation $\sqrt z$ usually refers to the principal square root. If your equations are interpreted with this convention, then the first of your two options is correct. However, there may be situations that require you to sum different roots of the same value. In that case, you would need a different notational device to represent the left-hand side, e.g., you could write

$\sum_{\omega^2=z}\omega\;,$

and in that case you might need something like your second option to compute this sum.

Your first equation is incorrect, since you get two different values on the right-hand side for different values of $k$, so you need some form of notation on the left-hand side that depends on $k$. For instance, you could say that the square roots of $z$ are given by

$\omega_k=\sqrt{r}\left(\cos(\varphi/2+k\pi)+\mathrm i\sin (\varphi/2+k\pi)\right)\;.$

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The sum of the $n$ complex roots of any polynomial $P_n(x)=x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$ is $(-a_1)$ by Vieta's formulæ. The $n$-th roots of $z$ are the $n$ roots of the polynomial $P_n(x)=x^n-z$ by definition hence their sum is zero for every $z$ and every $n\geqslant2$.

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    student: now that the dust settled, let me mention that I still do not know what was your question. I read it carefully and I thought I knew unambiguously what it was when I posted my answer but now, you have added comments which do not fit what I thought, nor, in my opinion, the text of your question. In the end, this is to suggest you try to reach more precise formulations of your future questions, if any.2011-10-23