Definition (Residue). Let be Holomorphic, and let . We define
where is a positively oriented simple closed curve enclosing .
Residue theorem
Definition: The Residue theorem states that the line integral of curve is the sum of it’s residues inside
Applications to computing integrals
Example 1. Consider
This is perhaps not the best example of the power of the Residue theorem, considering we know from elementary calculus that and therefore
However, the method we will describe does not rely on the fundamental theorem of calculus (namely, the statement that ), and therefore can be used to compute integrals in scenarios when the anti-derivative of a function cannot be expressed in terms of “elementary functions.” Here, elementary functions refer to the canon of functions we deem “elementary” (composition, sums, and products of functions like , , polynomials, rational functions, trig/hyperbolic functions).
As a first step, we note that is holomorphic wherever it is defined, and has simple poles (i.e. poles of order 1) at . Now, we just need to bring our integral into a setting in which the Residue theorem applies. We define a simple closed curve to be a semicircle in the upper-half plane with diameter running along the real axis. To be precise, we define for and define for . This curve will enclose the pole at for .
We have that
and for , we have by the residue theorem that
Now, let . The idea is that if then
Thus, we have reduced computing this integral to computing the residue of at . There are many methods for computing the residue of poles (see wikipedia). Here, I’ll compute the residue of at the simple pole as
Thus,
Details
Above is a good outline of the typical integral computation using residues. However, I skipped the proof that as , as well as the necessary steps to make the argument that rigorous.
By definition of the line integral,
It follows that
Hence,
It follows that we can say
Since the limits of the summands in the RHS converge,
We have that