How do I calculate $\displaystyle\int_{C_1(1)}(1+e^z)^{-1}\text{d}z$?
I have tried parametrizing $C_1(1)$ by $z=1+e^{i\theta}$ with $\theta \in [0,2\pi]$, but this does not help much as I would have a double exponential. Any ideas on how to progress are welcome.
That integral is equal to $0$, since the disk $D_\pi(1)$ is simply connected, the image of $C_1(1)$ is contained in it and the function$$\begin{array}{ccc}D_\pi(1)&\longrightarrow&\Bbb C\\z&\mapsto&\dfrac1{1+e^z}\end{array}$$is analytic.
I chose the disk $D_\pi(1)$ because, for each $z$ in that disk, $1+e^z\ne0$.