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# 10: Definite Integrals Using the Residue Theorem

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In this topic we’ll use the residue theorem to compute some real definite integrals.

$\int_{a}^{b} f(x)\ dx$

The general approach is always the same

1. Find a complex analytic function $$g(z)$$ which either equals $$f$$ on the real axis or which is closely connected to $$f$$, e.g. $$f(x) = \cos (x)$$, $$g(z) = e^{iz}$$.
2. Pick a closed contour $$C$$ that includes the part of the real axis in the integral.
3. 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.
4. Use the residue theorem to compute $$\int_C g(z)\ dz$$.
5. Combine the previous steps to deduce the value of the integral we want.