So I want to know how to evaluate $$\int_0^\infty f(x)dx$$for integrable $f(x)$ using the quarter circle contour, assuming that $f(x)$ has no singularities that aren't poles in that region. So, I know that by using residue theorem $$\int_C f(x)dx=2\pi i\sum_k\text{Res}(f,z_k)$$Where $z_k$ is a pole of $f$. So is the following true $$\int_C f(x)dx=\Re\left(\int_0^Rf(z)dz+\int_0^{Ri}f(z)dz+\int_\Gamma\right)$$(Where $\Gamma$ is the arc)? I don't really understand how to use the quarter circle contour to evaluate integrals. So is my thinking right? If not, how am I supposed to use this contour? As a note, I got $Ri$ since there is a radius point upward from $0$ to $Ri$.
2026-04-07 19:29:37.1775590177
How to use Quarter-circle contour?
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