\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\un}{\underline}
\newcommand{\undb}{\underbrace}
\newcommand{\pl}{\partial}
\parindent=0pt
\begin{document}

{\bf Question}

What is the image of the wedge defined by $0 \leq \arg z \leq
\ds\frac{\pi}{n},\ n$ integer, under the transformation
$w=f(z)=z^m,\ m$ integer? Discuss any special values of $m$. What
happens if $n$ or $m$ is non-integer? What happens when $m>2n$?


\medskip

{\bf Answer}

Define wedge boundaries by $z=re^{i\frac{\pi}{n}}$ and $z=r,\
r>0$.

(z)

\setlength{\unitlength}{.5in}
\begin{picture}(6,2)
\put(0,0){\vector(1,0){6}}

\put(2,0){\vector(0,1){2}}

\put(2,0){\circle*{.125}}

\put(6,0){\makebox(0,0)[l]{$x$}}

\put(2,2){\makebox(0,0)[bl]{$y$}}

\put(2.75,.25){\makebox(0,0){$\frac{\pi}{n}$}}

\put(5,2){\makebox(0,0)[l]{$z=re^{i\frac{\pi}{n}}$}}

\put(2,0){\line(3,2){3}}
\end{picture}

Then $w=f(z)=z^m$ gives

$$w=r^me^{i m\frac{\pi}{n}},\ z=r^m$$

i.e., a wedge of angle of opening $\ds\frac{m\pi}{n}$:

(w)

\setlength{\unitlength}{.5in}
\begin{picture}(6,2)
\put(0,0){\vector(1,0){6}}

\put(2,0){\vector(0,1){2}}

\put(2,0){\circle*{.125}}

\put(6,0){\makebox(0,0)[l]{$u$}}

\put(2,2){\makebox(0,0)[bl]{$v$}}

\put(2,.25){\makebox(0,0)[l]{$\frac{m\pi}{n}$}}

\put(2,0){\line(-2,3){1.25}}
\end{picture}

Note that if $m=n$ we have wedge in $z \longrightarrow$ upperhalf
plane in $w$.

If $m=2n$ we have wedge in $z \longrightarrow$ complete $w$-plane.

If $m$ or $n$ is non-integer, we just have an irrational \lq\lq
fraction" of $\pi$ as an opening angle of the sector in (w).

If $m>2n$ we map the $z$-wedge onto more than one revolution of
the (w) plane.

\newpage
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\begin{picture}(3,2)
\put(0,0){\vector(1,0){3}}

\put(1,0){\vector(0,1){2}}

\put(1,0){\circle*{.125}}

\put(3,0){\makebox(0,0)[l]{$x$}}

\put(1,2){\makebox(0,0)[bl]{$y$}}

\put(3,2){\makebox(0,0)[bl]{$A$}}

\put(3,0){\makebox(0,0)[tr]{$B$}}

\put(1,0){\makebox(0,0)[tr]{$0$}}

\put(1.75,.25){\makebox(0,0){$\frac{\pi}{n}$}}

\put(1,0){\line(1,1){2}}
\end{picture} \hspace{.25in}
$\longrightarrow$\hspace{.25in} \setlength{\unitlength}{.5in}
\begin{picture}(4,4)
\put(0,0){\vector(1,0){4}}

\put(2,-2){\vector(0,1){4}}

\put(2,0){\circle*{.125}}

\put(4,0){\makebox(0,0)[l]{$u$}}

\put(2,2){\makebox(0,0)[bl]{$v$}}

\put(2.75,.25){\makebox(0,0)[l]{$\frac{m\pi}{n}>2\pi$}}

\put(2,0){\line(2,1){2}}

\put(4,1){\makebox(0,0)[bl]{$A'$}}

\put(4,0){\makebox(0,0)[tr]{$B'$}}

\put(2,0){\makebox(0,0)[tr]{$0$}}
\end{picture}

\vspace{2in}

This is bad, since it can lead to ambiguities, i.e., certain
values of $w$ refer to \un{two} values of $z$ (in the overlap
region of $w$-plane).
\end{document}
