Integral of $\frac{1+2z}{1+z}dz$

Question

Let $C$ be a curve in the upper half plane with initial point $-1+2i$ and final point $1+2i$.

Let $f(z) = \frac{1+2z}{1+z}dz$

I need to find the integral of $f$ over $C$.

If I take $C(t) = (1-2t) + 2i$ for $t \in [0,1]$ then I am getting some value for the integral.

But how do I prove that the integral is path independent?

Answer 1

That integral is path-independent because the upper half-plane is simply connected and the integral of any analytic function defined on a simply connected set along a path $\gamma$ only depends upon the initial and final points of $\gamma$.