The following is problem 19 from page 87 of Saff and Snider's "Fundamentals of Complex Analysis for Mathematics, Science, and Engineering,"
How would you construct a branch of $\log z$ that is analytic in the domain D consisting of all points in the plane except those lying on the half-parabola $\lbrace x+iy: x \ge 0, y = \sqrt{x}\rbrace$?
Saff and Snider have defined all logarithms to be taken to the base of $e$ unless otherwise mentioned. I understand the idea of a branch cut and its purpose in constructing a single-valued function from a multi-valued one. Saff and Snider have also defined the principal logarithm of $z$ as,
$ \text{Log}\;z = \text{Log}\;|z| + i\;Arg\; z $
Where $\text{Arg} \; z$ lies in the half-open interval $(-\pi,\pi]$. This function has a branch cut on the nonpositive real axis. So I thought of considering $\text{Log}\;z^2.$ I figured that substituting $z^2$ for $z$ might result in a branch cut that resembled a quadratic. Unfortunately, $\text{Log}\;z^2$ just has two branch cuts. They lie on the nonnegative and nonpositive imaginary axes, respectively. I also considered using a branch cut of $\log z$ whereby the arguement is taken to be on the half-open interval $(\pi/4,9\pi/4]$. This is close to the answer, but the branch cut is still in the shape of a ray rather than a half-parabola. I'm not sure what else to try...