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\title{ Complex Analysis Prelim. (August 25, 2005)}
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1. Find an analytic function $f(z)$ whose real part is
$$
Re(f(z))=xy+3, \ (z=x+iy).
$$
Does such a function exist? Justify your answer.

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2. Let $u$ be a real-valued harmonic function. For what functions $f$ is the
function $f(u)$ harmonic?

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3. Let $f$ be analytic in a domain $\Omega$ and $Re(f)$ be a constant on $\Omega $.
Show that $f$ is a constant.

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4. Construct a conformal mapping of ${\mathbb C}\setminus \left( [-1,0]\cup
 [-i,i] \right ) $ onto the init disk.

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5. State and prove Morera's Theorem.

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6. Compute the integral
$$
\int_{|z|=1/2} \frac{dz}{(2z-\bar{z})^8}.
$$

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7. How many roots of the equation $z^4-6z+3=0$ have their modulus between $1$ and$2$?

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8. Let $f$ be an entire function whose modulus is  constant on some circle.
Prove that $f(z)=C(z-z_0)^n$.

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9. By Picard's Theorem every meromorphic function has at most 2 exceptional
values (that is there are at most two complex numbers which are not in the
range). How many exceptional values does $\tan z$ have? Find them.

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10. Prove that there exists a constant $C$ such that for every polynomial $P$
$$
\left | \int_{-1/2}^{1/2} P(x)dx \right |\leq C\int_{|z|=1}|P(z)||dz|.
$$














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