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For what values of $m$ and $n$ does $\int_C z^mz^{-n}dz=0$ and for what values does $\int_C z^mz^{-n}dz=2i\pi?$

I am stuck on this problem, any hint?

Thanks

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Is $C$ supposed to be a circle around origin? Are $m, n$ integral? –  Marek Mar 14 '13 at 15:21
    
@Marek $C$ is a contour, nothing is specified about $m$ and $n$ –  Alti Mar 14 '13 at 15:24
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1 Answer

up vote 1 down vote accepted

Assuming $C$ is a unit circle around origin we have this. Parametrize the circle as $z = e^{it}$, $t \in [0, 2\pi]$. Then we have $$\int_C z^{m-n} dz = \int_0^{2\pi} e^{it(m-n)} (i e^{it}).$$ Now if $m - n + 1 = 0$, the integrand is constant and we get the result $2\pi i$. Otherwise the integral vanishes.

Now, for general curve, the integral is only nonzero when $m -n = -1$. In that case, it measures what is called a winding number $w$, i.e. how many times does the contour wind around the origin, and the result is $2 \pi i w$. We had a special case of this in the first paragraph where $w = 1$, since the circle wraps around the origin just once.

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That was helpful, thanks! –  Alti Mar 14 '13 at 15:47
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