Chapter 14.9, Problem 6P

For each of the following functions $w=f(z)=u+iv,$ find $u$ and $v$ as functions of $x$ and y. Sketch the graphs in the $(x,y)$ plane of the images of $u=$ const. and $v=$ const. for several values of $u$ and several values of $v$ as was done for $w=z^2$ in Figure 9.3. The curves $u=$ const. should be orthogonal to the curves $v=$ const.

$w=\sqrt{z}.$ Hint: This is equivalent to $w^2=z;$ find $x$ and $y$ in terms of $u$ and $v$ and then solve the pair of equations for $u$ and $v$ in terms of $x$ and y. Note that this is really the same problem as Problem 1 with the $z$ and $w$ planes interchanged.