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QUESTION


Evaluate $\int_C z^m\bar z^n dz$\ where $C$ is the unit circle
$|z|=1$, and $m$ and $n$ are integers.



ANSWER


If we parametrize the unit circle $C$ as $z=e^{it}$, $(0\le t \le
2\pi)$, then we get $\int_Cz^m\bar z^n
dz=i\int_0^{2\pi}e^{(m+1-n)it}dt={i\over
(m+1-n)i}[e^{(m+1-n)it}]_0^{2\pi} =0$ if $m+1\not =n$, and if
$m+1=n$ we get $2\pi i$.




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