In Schaum's complex variable book, there is an exercise in contour integration: $$ \int \overline{z}^{2} dz $$ over $|z|=1$.
The answer seems to be $0$, but when I integrate like this using contour integration formula, $$ \int_{0}^{2\pi}e^{-2i\theta} i e^{i\theta} d\theta $$ $$ -e^{-2i\pi}. $$ Then I get $-1$ as a result. :(
What am I doing wrong?
$$ \int_{0}^{2\pi}e^{-2i\theta}\,i\,e^{i\theta}\,d\theta=i\int_{0}^{2\pi}e^{-i\theta}\,d\theta=i\,\frac{e^{-i\theta}}{-i}\,\Bigr|_0^{2\pi}=0. $$