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\centerline{\sectionfont Final Examination Math 605, Christmas
2002 }
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{\bf Instructions} All work should be on the 10 pages supplied. Write in your
answer after you are sure it is your best effort. Turn in your completed script
to Carole in Room 255 by 4pm Wednesday, December 18.
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1. Verify that $u(x,y)=e^x(x\cos y-y\sin y)$ is a harmonic
function on
$\Bbb C$. Find an analytic function $f(z)$ on $\Bbb C$ such that
$u(x,y)=Re f(z)$. You should write $f(z)$ explicitly as a function of $z$
(with no $x,y,\bar z$ or integrals in the expression).
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(1 Continued)
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2. Consider the region $\Omega= \Bbb C-\{z\in \Bbb C|Re z\ge 0\}$. Each point in
this region can be given unique polar coordinates $(r,\theta)$ with $r>0$ and
$0<\theta<2\pi$. Show that $u(r,\theta)=\theta$ defines a harmonic function on
$\Omega$. What is "the" conjugate harmonic function to $u$ and what is the largest
region on which it can be defined.
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(2 Continued)
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3. Suppose $f$ is a non-constant analytic function on $\Bbb C$ which is bounded on
the upper half-plane $\Bbb H$. What kind of isolated singularity is the
point $\infty$ of the function $f$ --- removable, pole or essential? Explain your
answer. Give an example of such a function.
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(3 Continued)
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4. Show that there are no non-constant non-negative harmonic functions on $\Bbb
R^2$. Can a non-constant harmonic function on $\Bbb
R^2$ not take the value 2002.
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(4 Continued)
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5. Let $f$ be a continuous complex-valued function on the closed unit disc $\bar
D$ which is analytic in $D$. If $f=constant$ along an arc of $\partial D$ show
that $f\equiv constant$ on $D$.
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(5 Continued)
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