Complex contours and residues

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

For certain real integrals extended to complex functions, choose a contour and compute enclosed poles. Justify the arc contribution, orientation and analytic hypotheses. This complex-analysis technique is not an automatic calculator method.

Step-by-step example

1. Close 1/(z²+1) in the upper half-plane.

   $$f(z)=\frac1{(z-i)(z+i)}$$

2. The only enclosed pole is i.

   $$\operatorname{Res}(f,i)=\frac1{2i}$$

3. The arc vanishes as its radius grows.

   $$\int_{-\infty}^{\infty}\frac{dx}{1+x^2}=2\pi i\frac1{2i}=\pi$$