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QUESTION


Evaluate (the complex integral) $\int_{1}^2 {z^kdz}$, where $k >
-1$ is an integer. Explain why it makes sense to evaluate such an
integral. What happens if $k=-1$? What happens if $k<-1$?



ANSWER


It makes sense to evaluate $\int_1^2z^k dz$ as long as the
integral is independent of the path. This is the case if
$\int_{\gamma}z^kdz=0$ around any closed path ${\gamma}$. This is
true if $z^k$ has an antiderivative which is the case if
$k\not=-1$. (If $k=-1$ then we are not allowed to use Log$z$ as an
antiderivative as Log is not analytic in a neighbourhood of 0.) If
$k\not=-1$ then $\int_1^2z^k dz={1\over
k+1}[z^{k+1}]_1^2={2^{k+1}-1\over k+1}$.




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