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Could someone help me through this problem?

This theorem is already proved: Suppose f is entire and D is the boundary of a rectangle R. Then $\displaystyle\int_{D} f(z)\, dz$

Now he must prove this directly from the theorem: given any rectangle with vertices (a, c), (b, c), (b, d) and (a, d), parameterize the boundary D and verify directly that $\displaystyle\int_{D}\, dz=$$\displaystyle\int_{D}z\, dz=0$

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    I agree with copper.hat: the point of this exercise would be to verify the theorem *by hand*, not by applying the theorem. That is: parameterize $D$ and calculate the two integrals and check that they are both zero.2012-04-24

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$f(z)=z-1$ is entire, so $\int_D f(z)dz=0$. This translates to $\int_D dz=\int_D z dz$.

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    Apply the same result for $f(z)=z+1$.2012-04-24