Suppose the function $f(z)$ has singularities and I want to calculate the integral $$\int_{-\infty}^{\infty}f(z)dz=?$$
I use Residue theorem
$$\oint_cf(z)dz=2\pi i\sum\text{Res}(f,z_0)$$
And
$$\oint_cf(z)dz=\int_{-\infty}^{\infty}f(z)dz+\int_{arc}f(z)dz$$
Quenstion: In general, is the arc part always zero? How to calculate the arc part?
$$\int_{arc}f(z)dz=?$$
If necessary, pls show a specific example.