10: Definite Integrals Using the Residue Theorem
- Page ID
- 6535
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In this topic we’ll use the residue theorem to compute some real definite integrals.
\[\int_{a}^{b} f(x)\ dx \nonumber \]
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. \(f(x) = \cos (x)\), \(g(z) = e^{iz}\).
- 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 \(\int g(z)\ dz\) over each of the pieces except the part on the real axis.
- Use the residue theorem to compute \(\int_C g(z)\ dz\).
- 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.