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\textbf{Math 425: Complex Variables}
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\textbf{Quiz 6}\quad \textsl{March 8, 2002}

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1. Let $\gamma(t) = t e^{it}$, $0 \le t \le 4\pi$:
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Compute $\int_\gamma e^z\,dz$.

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2. Use Cauchy's Formula to find the value of $\int_{|z|=2}
\frac{z}{z^2+1}\,dz$.


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3. Let $f$ be analytic on a domain $D$ and let $C$ be a circle $z_0 +
re^{it}$, $0\le t \le 2\pi$, contained in~$D$. Prove that the average
value of $f$ on $C$ equals the value of $f$ at the center of $C$,
\[ f(z_0) = \frac{1}{2\pi}\int_0^{2\pi} f\big(z_0+re^{it}\big)\,dt \]

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