Find $\int _C e^z dz$, where $C$ is any path from 0 to 1 in $\Bbb C$.
It seems pretty easy, while, for a straight line path, I can simply calculate $e^1-e^0$ which I got $e-1$, while if the path is an unit circle, then \begin{align} \int_C e^z dz&=\int_{0}^{0} e^{e^{i\phi}}ie^{i\phi}d\phi\\\\ &=\left. e^{e^{i\phi}}\right|_{0}^{0}\\\\ &=e^{e^0}-e^{e^{0}}\\\\ &=0 \end{align} since 1 and 0 both at angle 0 degree. How did this conflict generated? Is there any other conditions I need to point out for "any path"?
By Cauchy's Theorm the integral is same for all paths from $0$ to $1$. The unit circle is not a path from $0$ to $1$.