Consider the integral
$$I=\int _\Gamma\frac{1}{4+i(\log z)^2}dz$$
Where $\Gamma$ encircles the unit circle infinitely many times.
Would it then make sense to use a parameter n: encirclement count, $z=e^{i t}$ $\log z = i (t+2 \pi n)$
$$I[n]=\int^\pi _{-\pi}\frac{1}{4-i(t+2 \pi n)^2}i e^{it}dt$$
$$I=\sum ^ \infty _{n=-\infty}I[n]$$