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\textbf{Math 425: Complex Variables}
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\textbf{Quiz 1}\quad \textsl{January 30, 2004}

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1. Find the real and imaginary parts of
$\Big(\frac{2-z}{2-\overline{z}}\Big)^{15}$ when $z = 1 - i\sqrt{3}$.

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2. Show that if $2 < |z| < 3$, then $\Big|\frac{z-3}{z^2+4z-5}\Big| < 3$.

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3. Let $S$ be an arbitrary subset of the plane. Show that every point of
$S$ is either an interior point or a boundary point of $S$.

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4. a) Write the function $f(z) = z^4 - z^2$ in the form $f(x+iy) = u(x,y) +
iv(x,y)$.

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b) Write $f(x+iy) = \frac{x^3+xy^2+x}{x^2+y^2} +
i\frac{y^3+x^2y+y}{x^2+y^2}$ as a function $f(z)$ of $z = x+iy$ (in
simplest possible form).

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5. Find the limit $\lim_{z\to a}\frac{z^2 - \sqrt{2}z + 1}{z^4 + 1}$ where
$a = (1+i)/\sqrt{2}$.

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