Sunday, May 11, 2014

Complex Analysis: #20 Symmetric Real Functions

  • Complex Analysis: #20 Symmetric Real Functions

For example, how do we calculate the integral

Complex Analysis: #20 Symmetric Real Functions equation pic 1

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

Complex Analysis: #20 Symmetric Real Functions equation pic 2

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

Complex Analysis: #20 Symmetric Real Functions equation pic 3

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  ∫0f(x)dx, using the Cauchy principle value technique.

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