# Complex contour integration

• Apr 2nd 2010, 04:56 PM
Eudaimonia
Complex contour integration
Let $\gamma$ be the square with vertices 0, 1, $\imath+1$, $\imath$ traversed counterclockwise. Evaluate
$\int_{\gamma}{|z|^2dz}$.

I parametrized and then deduced the formula:
$\int _{\gamma }\left|z|^2dz\right.=\sum _{i=1}^4 \int _{\gamma _i}|\gamma _i(t)|^2\gamma _i'(t)dt$. I come up with -1+ $\imath$ as the answer. Please verify, thank you.
• Apr 3rd 2010, 11:14 AM
Opalg
Quote:

Originally Posted by Eudaimonia
Let $\gamma$ be the square with vertices 0, 1, $i+1$, $i$ traversed counterclockwise. Evaluate
$\int_{\gamma}{|z|^2dz}$.

I parametrized and then deduced the formula:
$\int _{\gamma }\left|z|^2dz\right.=\sum _{i=1}^4 \int _{\gamma _i}|\gamma _i(t)|^2\gamma _i'(t)dt$. I come up with $-1+i$ as the answer. Please verify, thank you.

• Apr 3rd 2010, 03:49 PM
davismj
Quote:

Originally Posted by Opalg

Why wouldn't the answer be zero under Cauchy's theorem?
• Apr 3rd 2010, 04:36 PM
Eudaimonia
Please correct me if I'm wrong, but I don't think f(z)=|z|^2 is holomorphic at 0, and thus it does not meet that requirement of Cauchy's integral theorem.
• Apr 3rd 2010, 04:43 PM
xxp9
because |z|^2 = zz* is not analytic
Quote:

Originally Posted by davismj
Why wouldn't the answer be zero under Cauchy's theorem?

• Apr 3rd 2010, 05:21 PM
davismj
Quote:

Originally Posted by xxp9
because |z|^2 = zz* is not analytic

duh