1. ## complex integral

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

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

Is this correct?

2. I would parametrise the contour by $y(t)=it, 0\leq t\leq 1$

Then $I=\int_0^1\frac{e^{it}}{1+e^{it}}i~dt$
$=\int_0^1 i-\frac{i}{1+e^{it}}dt$
$=\left[ it-ln(1+e^{it}) \right]_0^1$
$=i-ln(1+e^i)+ln(2)$

Not sure if that's correct but it makes more sense to me.