Evaluate $\displaystyle \int_{C_{z}} \frac{e^{z} \ dz}{1+e^{z}} $ where $\displaystyle C_{z} $ is that piece of the imaginary axis lying between $\displaystyle y = 0 $ and $\displaystyle y = 1 $ and traversed upwards.

So let $\displaystyle w = e^z $. Then $\displaystyle C_{w}: |w| = 1, \ \ 0 \leq \text{arg}(w) \leq 1 $ traversed counterclockwise. So $\displaystyle I = \int_{C_{w}} \frac{dw}{1+w} $. Then we get $\displaystyle I = \text{Log}(1+e^{i})- \ln 2 $.

Is this correct?