• Complex Analysis: #20 Symmetric Real Functions

For example, how do we calculate the integral

On the one hand, there is no singularity at 0, but on the other hand, how are we to calculate the integral other than by using the calculus of residues? [For example, partial integration only seems to make things more complicated here. Nevertheless, I see that the computer algebra system MuPAD does give the correct answer with little fuss!] Let us begin by noting that

But the residue of exp(ix)/x at 0 is simply 1. Therefore, theorem 38 shows that

This example shows that if we have a meromorphic function f which is defined in ℂ, such that f(x) = f(−x) for all x ∈ ℝ, and such that the conditions of theorem 38 are satisfied, then it does make sense to calculate the integral  ∫₀^∞ f(x)dx, using the Cauchy principle value technique.