91Ó°ÊÓ

Find three sets in the complex plane that map onto the set \(\arg (w)=\pi\) under the mapping \(w=z^{3}\).

Find four sets in the complex plane that map onto the circle \(|w|=4\) under the mapping \(w=z^{4}\).

Do lines that pass through the origin map onto lines under \(w=z^{n}, n \geq 2 ?\) Explain.

(a) Is it true that \(\lim _{z \rightarrow z_{0}} \overline{f(z)}=\lim _{z \rightarrow z_{0}} f(\bar{z})\) for any complex function \(f ?\) If so, then give a brief justification; if not, then find a counterexample. (b) If \(f(z)\) is a continuous function at \(z_{0}\), then is it true that \(\overline{f(z)}\) is continuous at \(z_{0}\) ?

If \(f\) is a function for which \(\lim _{x \rightarrow 0} f(x+i 0)=0\) and \(\lim _{y \rightarrow 0} f(0+i y)=0\), then can you conclude that \(\lim _{z \rightarrow 0} f(z)=0 ?\) Explain.

Find the image of the half-plane \(\operatorname{Im}(z) \geq 0\) under each of the following principal \(n\) th root functions. (a) \(f(z)=z^{1 / 2}\) (b) \(f(z)=z^{1 / 3}\) (c) \(f(z)=z^{1 / 4}\)

Find the image of the region \(|z| \leq 8, \pi / 2 \leq \arg (z) \leq 3 \pi / 4\), under each of the following principal \(n\) th root functions. (a) \(f(z)=z^{1 / 2}\) (b) \(f(z)=z^{1 / 3}\) (c) \(f(z)=z^{1 / 4}\)

Consider the multiple-valued function \(F(z)=z^{1 / 3}\) that assigns to \(z\) the set of three cube roots of \(z\). Explicitly define three distinct branches \(f_{1}, f_{2}\), and \(f_{3}\) of \(F\), all of which have the nonnegative real axis as a branch cut.

Find a function that maps the entire complex plane onto the set \(2 \pi / 3<\arg (w) \leq 4 \pi / 3\)

Consider the multiple-valued function \(F(z)=(z-1+i)^{1 / 2}\). (a) What is the branch point of \(F ?\) Explain. (b) Explicitly define two distinct branches of \(f_{1}\) and \(f_{2}\) of \(F .\) In each case, state the branch cut.