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\textbf{Math 425: Complex Variables}
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\textbf{Quiz 2}\quad \textsl{February 13, 2004}

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\begin{problem}{1.}%
Use the definition of the derivative to prove $\frac{d}{dz}
\Big(\frac{1}{z^2+1}\Big) = -\frac{2z}{(z^2+1)^2}$ for
$z\in\C\setminus\{i,-i\}$.
\end{problem}

\begin{problem}{2.}%
Calculate $\frac{\partial f}{\partial z}$, $\frac{\partial f}{\partial
\overline{z}}$, and $\frac{df}{dz}$ (if it exists) for
$f(x+iy) = x^3 + xy^2 - i(x^2y + y^3)$.
\end{problem}

\begin{problem}{3.}%
Define the complex function $e^z$ and show that $\frac{d}{dz}e^z = e^z$.
\end{problem}

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\begin{problem}{4.}%
Show that $u = x^3 - 3xy^2 - 2xy$ is harmonic, find its harmonic conjugate
$v$, then write $u + i v$ as a function of $z = x+iy$.
\end{problem}


\begin{problem}{5.}%
Use the definition of the complex function $\cos(z)$ to establish the identity:
\[ \cos^2(z) = \frac{1+\cos(2z)}{2} \]
\end{problem}

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