If X is path connected, all the groups $\pi_1(X, x)$ are isomorphic, so it is tempting
to try to “identify” all these groups with one another and to speak simply of the fundamental
group of the space X, without reference to base point. The difficulty with
this approach is that there is no natural way of identifying $\pi_1(X, x0) $with $\pi_1(X, x1)$;
different paths α and β from x0 to x1 may give rise to different isomorphisms between
these groups. For this reason, omitting the base point can lead to error.
I would like to add some commentary to the excellent answer by @DrMV,
showing how to compute the residues involved. We will use
$$f(z) = \frac{1}{1+z^2}
\exp(1/2 \times\mathrm{LogA}(1+z))
\exp(1/2 \times\mathrm{LogB}(1-z)).$$
Here $\mathrm{LogA}$ denotes the branch of the logarithm where $-...
