Friday, March 3, 2017

Properties of the principal square root of a complex number



I am studying the principal square root function of complex numbers. On Wikipedia they present a complex number $z$ using polar coordinates as



\begin{equation}
z = r \mathrm{e}^{i \varphi}, \quad r \ge 0, ~ -\pi < \varphi \le \pi.
\end{equation}



Further, they define the principal square root of $z$ as




\begin{equation}
\sqrt{z} = \sqrt{r} \mathrm{e}^{i \varphi/2}. \tag{1}
\end{equation}



Continuing, it is mentioned that




The principal square root function is thus defined using the
nonpositive real axis as a branch cut. The principal square root

function is holomorphic everywhere except on the set of non-positive
real numbers (on strictly negative reals it isn't even continuous).




I do not understand these two statements. My questions are




  1. Why is the principal square root function defined using the nonpositive real axis as a branch cut? It seems to me that for $z = \mathrm{e}^{i \pi}$, we obtain by equation $(1)$ the principal square root $\sqrt{z} = \sqrt{1} \mathrm{e}^{i \pi/2} = i$.

  2. Why is the principal square root function not continuous on the negative reals?



Answer



That is a convention that for principal value in general we must use $-\pi<\arg z<\pi$ for other values you can change the branch cut.


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