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{\bf Question}

Define the logarithm of a non-zero complex number and discuss its
multi-valued nature.  If $a,b\in{\bf C}$ and $a\not=0$ define
$a^b$.  Show that $a^i$ has infinitely many values.

Show how $z^i$ changes its value by choosing a particular value
$w_0$ of $z_0^i$ and then passing once around the circle with
centre $o$ and radius $|z_0|$.  Explain how you would define a
single-valued branch $f(z)$ of $z^i$ which gives an analytic
function in the upper half plane $U$.  If $r=|z|, \,\, \theta=\arg
z$ calculate for the particular branch you have chosen, $|z^i|$
and $\arg z^i$.

Find the image $f(u)$ of the upper half plane under $f$.  Also,
find the images of the following subsets of $U$.

\begin{itemize}
\item[i)]
$\{z\in U|\,|z|=r_1\}$


\item[ii)]
$\{z\in U|\arg z=\theta_1\}$.

\end{itemize}


\vspace{0.25in}

{\bf Answer}

If $z=e^w$ then we say that $w$ is a logarithm of $z$.  Now $e^w$
is periodic with period $2\pi i$, so $z=e^{w+2n\pi i}$, and
$w\not=2n\pi i$ is also a logarithm of $z$, for $n\in{\bf Z}$.

Now let $w=x+iy$ so $z=e^xe^{iy}$ thus $e^x=|z|$ and $y=\arg z$.

So $w=\ln|z|+i\arg z$, and again the multi-valued nature is clear
from that of $\arg z$.

We define $a^b$ by $a^b=\exp(b\log a)$

So $a^i=\exp(i\log a)=\exp(i[\log|a|+i({\rm Arg}a+2n\pi)])$

$=\exp i\ln|a|\exp(-{\rm Arg}a)\exp(-2n\pi) \hspace{0.2in}
n\in{\bf Z}$

The first two factors are uniquely defined and non-zero.  The
third factor is different for each $n\in{\bf Z}$.

Let $z$ pass round the circle centre $O$, radius $|z_0|$. Choosing
the value of $z^i$ with $n=0$ we have

$z^i=\exp(i\ln|z_0|)\exp(-{\rm Arg}z)=K\exp(-{\rm Arg}z)$

so with the principal value of Arg$z$ satisfying $-\pi\leq{\rm
Arg}z<\pi$, $z^i$ changes as follows:

when $z=-|z_0|$, i.e. Arg$z=-\pi, \,\, z^i=Ke^\pi$ as $z$ moves
round the circle Arg$z$ increases and so $z^i$ assumes values of
the form $Ke^{-\pi t} \hspace{0.2in} -1\leq t<1$.

As Arg$z\to\pi, \hspace{0.2in} z^i\to Ke^{-\pi}$, so there is a
discontinuity at $-|z_0|$ of magnitude $K(e^\pi-e^{-\pi})$.  To
choose a single-valued branch analytic for im$z>0$ the above will
do.

So if $r=|z|$ and $\theta={\rm Arg}z$ then $z^i=\exp(i\log
r)\exp(-\theta)$

So $|z^i|=e^-\theta, \hspace{0.2in} \arg(z^i)=\log r$

If $U$ is the upper half plane, $0<\theta<\pi$ and $r>0$, so $\log
r$ takes all real values and $e^{-\theta}$ takes values between 1
and $e^{-\pi}$.  So the image of $U$ is the open annulus
$e^{-\pi}<|w|<1$.

\begin{itemize}
\item[i)]
$\{z\in U|\,|z|=r_1\}$ - the subset of this annulus satisfying
$\arg w=\log r_1$.  i.e. a line segment.


\item[ii)]
$\{z\in U|\arg z=\theta_1\}$. - a circle of radius $e^{-\theta}$.

\end{itemize}


DIAGRAM


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