We set $C = \exp\left(\frac{I}{2\pi}\right)$ with:
$$I = \int_0^{2\pi} \ln\left(\max\left\{\left|e^{i\theta} - 1\right|, \left|e^{i\theta} + 1\right|\right\}\right) d\theta$$
Show that:
$$I = 4\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}$$
You may use the result from question 2.13.