Example 6. Compute the integral

$$
I=\int_{|z|=2} \frac{d z}{1+z^{4}}
$$
The poles (finite) of the integrand

$$
f(z)=\frac{1}{1+z^{4}}
$$

are the roots $z_{1}, z_{2}, z_{3}, z_{4}$ of the equation $z^{4}=-1$, which all lie inside the circle $|z|=2$. The function $f(z)=\frac{1}{1+z^{4}}$ has an expansion in the neighborhood of the infinitely distant point

$$
f(z)=\frac{1}{1+z^{4}}=\frac{1}{z^{4}} \frac{1}{1+\frac{1}{z^{4}}}=\frac{1}{z^{4}}-\frac{1}{z^{8}}+\frac{1}{z^{12}}-\cdots
$$

from which it is clear that res $f(\infty)=-c_{-1}=0$. By the equality (3)

$$
I=2 \pi i \sum_{k=1}^{4} \operatorname{res} f\left(z_{k}\right)=-2 \pi i \text { res } f(\infty)=0
$$