10: Definite Integrals Using the Residue Theorem
( \newcommand{\kernel}{\mathrm{null}\,}\)
In this topic we’ll use the residue theorem to compute some real definite integrals.
∫baf(x) dx
The general approach is always the same
- Find a complex analytic function g(z) which either equals f on the real axis or which is closely connected to f, e.g. f(x)=cos(x), g(z)=eiz.
- Pick a closed contour C that includes the part of the real axis in the integral.
- The contour will be made up of pieces. It should be such that we can compute ∫g(z) dz over each of the pieces except the part on the real axis.
- Use the residue theorem to compute ∫Cg(z) dz.
- Combine the previous steps to deduce the value of the integral we want.
- 10.1: Integrals of functions that decay
- The theorems in this section will guide us in choosing the closed contour C described in the introduction.