$y= | \sqrt x |$, should the graph extend to include complex numbers?

Question

Hey I have been wondering recently what the graph of $y=| \sqrt x|$ would look like if it was extended to complex numbers. I have been doing some research and it turns out that the absolute value of complex numbers does turn out to be a real number which would mean that this function would be graphable even in the negative numbers. I am just asking if anyone would know what this graph would look like or if they know where I could see it. Thanks!

Answer 1

Because the complex square root is not a function but a "double-valued function", it's not a priori obvious that $y = |\sqrt{x}|$ is a real-valued function. It turns out, however, that the two complex square roots of each complex number $x$ have the same magnitude, and that this magnitude is the non-negative real number $\sqrt{|x|}$. (If $x = \rho e^{i\theta}$ in polar form, then $\sqrt{x} = \pm \sqrt{\rho} e^{i\theta/2}$.) The graph is obtained by revolving the graph $y = \sqrt{x}$ (with $x \geq 0$) about the $y$-axis.

The diagrams below show the graph in Cartesian and polar coordinates, respectively: