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Math. 313, Complex Variables \\
Practice Exam I ---- Spring 2004 \\[.2in]
V. Coutsias \\[.2in]
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\begin{enumerate}

	\item (15pts.)
\begin{enumerate}
	\item (5pts) For which complex numbers does
\[ \left| e^{-iz}\right| < 1   \]
hold?
	\item (10pts) Find all values of 
\[ \sqrt{3+4 i} + \sqrt{3-4 i}      \ \ .\ \ \]
\end{enumerate}
          
	\item (10pts.) Show that the function $f(z) = \bar{z}$ is 
nowhere differentiable.

	\item (20pts.) Find all values of $z$ such that
\begin{enumerate}
\item $e^z = 1 + i\sqrt{3}$.
\item $\tan{z} = 2i $.
\item $z^i = 1$.
\item $\log{\left(i+\sqrt{z^2+3}\right)} = -\frac{\pi}{2}i$.
\end{enumerate}
 

	\item (15pts.) Show that $u(x,y) = \log{\sqrt{x^2+y^2}}$ is
harmonic in some domain (which?) and find a harmonic conjugate $v(x,y)$.

	\item (10pts.) Find a linear fractional transformation that maps the
points $\left\{0,1,i\right\}$ to the points $\left\{1,0,\infty\right\}$. 

	\item (10pts.) Show that 
\[\lim_{z\rightarrow \infty} \frac{z^2 + 1}{z-1} = \infty \]
 by using the equivalence
\[ \lim_{z\rightarrow \infty} f(z) = \infty \Longleftrightarrow
\lim_{z\rightarrow 0} \frac{1}{f\left(\frac{1}{z}\right)} = 0 \ . \]

	\item (10pts.) Find the angle of rotation and stretching factor
produced by the mapping $w = 1/z^2$ at $z=1$. 

	\item (10pts.) Sketch the region onto which the sector
$r\le 1\ ,\ 0\le \theta \le \pi/4$ is mapped by the transformation
$w = z^3$.
\end{enumerate}
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