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\centerline{\bf MATH 530 Qualifying Exam}
\centerline{January 1998}
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\centerline{Notation: $D_r(a)$ denotes the disk, $\{z\in\C:|z-a|<r\}$.}
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\item"{\bf1.}"
{\it (15 pts)}
Evaluate the integral\ \ $\ds{\int_{-\infty}^0\frac{x^2}{x^4+x^2+1}\,dx}$.
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\item"{\bf2.}"
{\it (15 pts)}
Find a one-to-one analytic map from $D_1(0)\cap\{x+iy\,:\,x,y>0\}$
onto $D_1(0)$.
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\item"{\bf3.}"
{\it (25 pts)}
Let $\Cal F$ denote the set of
analytic functions $f$ on $D_1(0)$ such that
$|f(z)|<1$ for all $z\in D_1(0)$, $f(0)=0$, and $f'(0)=0$.
Prove that if $f\in\Cal F$, then
$|f(z)|\le|z|^2$ for all $z\in D_1(0)$.  
Let $M=\sup\{|f''(0)|\,:\,f\in\Cal F\}$.   
Find all functions, if any, in $\Cal F$ such that $|f''(0)|=M$.
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\item"{\bf4.}"
{\it (15 pts)}
How many zeroes does the polynomial
$$z^{1998}+z+2001$$
have in the first quadrant?  Explain your answer.
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\item"{\bf5.}"
{\it (15 pts)}
Prove that a harmonic function cannot have an isolated zero.
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\item"{\bf6.}"
{\it (15 pts)}
Let $C_1(0)$ denote the unit circle $\{z\in\C\,:\,|z|=1\}$ and
let $f$ be a function that is analytic on $D_r(0)$ for some $r>1$.
Prove that if $f(C_1(0))\subset C_1(0)\setminus\{1\}$, then $f$ is
a constant function.

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